Published as Chapter 9 in: Handbook of Detergents, Part A: Properties, G. Broze, Ed.; M. Dekker, New York, 1999; pp. 303-418

EQUILIBRIUM AND DYNAMICS OF SURFACTANT ADSORPTION

MONOLAYERS AND THIN LIQUID FILMS by

Krassimir D. Danov, Peter A. Kralchevsky and Ivan B. Ivanov Faculty of Chemistry, University of Sofia, Sofia, BULGARIA

TABLE OF CONTENTS

Part 1 (http://www.lcpe.uni-sofia.bg/publications/1999/pdf/1999-10-1-KD-PK-II.pdf) I. Introduction ………………………………………………………………………………….. 1 II. Thermodynamics of adsorption from surfactant solutions …………………….…………… 3 A. Langmuir adsorption isotherm and its generalizations B. Van der Waals type adsorption isotherms III. Kinetics of adsorption from surfactant solutions ……………...…………………………… 9 A. Basic equations B. Adsorption under diffusion control C. Adsorption under barrier control D. Electro-diffusion control (adsorption of ionic surfactants) E. Adsorption from micellar solutions F. Rheology of adsorption monolayers IV. Flexural properties of surfactant adsorption monolayers ……………………..………….. 36 A. Mechanical description of a curved interface B. Thermodynamics of adsorption monolayers and membranes C. Interfacial bending moment of microemulsions D. Model of Helfrich for the surface flexural rheology E. Physical importance of the bending moment and the curvature elastic moduli V. Thin liquid films stabilized by surfactants ……………………………………….……….. 50 A. Thermodynamic description of a thin liquid film. B. Contact line, contact angle and line tension C. Contact angle and interaction between surfactant adsorption monolayers D. Films of uneven thickness

Part 2 (http://www.lcpe.uni-sofia.bg/publications/1999/pdf/1999-10-2-KD-PK-II.pdf) VI. Surface forces in thin liquid films ……………………………………….………………. 69 A. DLVO theory B. Hydration repulsion and ionic-correlation force C. Oscillatory structural and depletion forces D. Steric polymer adsorption force E. Fluctuation wave forces F. Surface forces and micelle growth VII. Hydrodynamic forces in thin liquid films ……………………………………………… 99 A. Films of uneven thickness between colliding fluid particles B. Inversion thickness and dimple formation C. Effect of surfactant on the rate of film thinning D. Role of surfactant transfer across the film surfaces VIII. Influence of surfactant on capillary waves …………………………………………… 116 A. Waves in surfactant adsorption monolayers B. Waves in surfactant stabilized liquid films IX. Summary ……………………………………………………………………...……. 129

References …………………………………………………………….……………. 131

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Part 1

I. INTRODUCTION

The interactions across a thin film, called the surface forces, to a great extent are determined by the surfactant adsorption at the film surfaces. From a physical viewpoint, a liquid film formed between two phases is termed "thin" when the interaction of the phases across the film is not negligible. Thin films appear between the bubbles in foams, between the droplets in emulsions, as well as between the particles in suspensions. Furthermore, the properties of the thin liquid film determine the stability of various colloids. There are three scenarios for the behavior of two colliding particles in a dispersion (for example, emulsion) depending on the properties of the films (Fig. 1). (i) When the film formed upon particle collision is stable, flocks of attached particles can appear. (ii) When the attractive interaction across the film is predominant, the film is unstable and ruptures; this leads to coalescence of the drops in emulsions or of the bubbles in foams. (iii) If the repulsive forces are predominant, the two colliding particles will rebound and the colloidal dispersion will be stable. In some cases, by varying the electrolyte concentration or pH, it is possible to increase the repulsion between the particles in a flocculated dispersion and to cause the inverse process of peptisation [1].

Fig. 1. Scenarios for the behavior of colliding emulsion droplets in connection with the droplet-droplet interaction, which is influenced by the properties of the surfactant adsorption monolayers.

2

The gap between two particles, like those depicted in Figure 1, can be considered as a

liquid film of uneven thickness. The calculation of the particle-particle interaction energy can

be reduced to the simpler calculation of the interaction across a plane-parallel film by using

the Derjaguin formula, see Eq. (182) below. This is the reason why in this review we will pay

much attention to the plane parallel films.

As already mentioned, the surfactants are used to stabilize the liquid films in foams,

emulsions, on solid surfaces, etc. Below we first consider the equilibrium and kinetic

properties of surfactant adsorption monolayers. Various two-dimensional equations of state

are discussed. The kinetics of surfactant adsorption is described in the cases of diffusion and

barrier control. Special attention is paid to the process of adsorption from ionic surfactant

solutions. Theoretical models of the adsorption from micellar surfactant solutions are also

presented. The rheological properties of the surfactant adsorption monolayers, such as

dilatational and shear surface viscosity, and surface elasticity are introduced. The specificity

of the proteins as high molecular weight surfactants is also discussed.

Next we consider the flexural properties of surfactant adsorption monolayers, which

are important for the formation of small droplets, micelles and vesicles in the fluid

dispersions. The contributions of various interactions (van der Waals, electrostatic, steric) into

the interfacial bending moment and the curvature elastic moduli are described. The effect of

interfacial bending on the interactions between deformable emulsion droplets is discussed.

Having reviewed the properties of single adsorption monolayers, we proceed with the

couples of interacting monolayers: the thin liquid films. First we present the thermodynamics

of thin films, and then we describe the molecular theory of the surface forces acting in the

thin films. We do not restrict ourselves to the conventional DLVO forces [2,3], but consider

also the variety of the more recently discovered non-DLVO surface forces [4]. The importance

of the micelle-micelle interaction for the mechanism of micelle growth is also discussed.

In addition to the surface forces, hydrodynamic forces play an important role for the

interactions in the real liquid films and colloidal dispersions. The hydrodynamic force is due

to the viscous friction accompanying the expulsion of the liquid from the gap between two

particles. When the particles are fluid (drops, bubbles) the fluidity of their surfaces

(determined by the properties of the surfactant adsorption monolayers) can significantly affect

the magnitude of the hydrodynamic force [5]. We consider the effect of the surface elasticity,

surface diffusion and surface viscosity on the hydrodynamic force. All these effects are

related to dilatational or shear deformations in the surfactant adsorption monolayer coupled

3

with the fluid fluxes in the bulk. Another effect of practical importance for emulsion

preparation is the influence of diffusion transfer of surfactant across the film surfaces on the

stability of the films. This effect is typical for emulsion systems, for which the surfactant is

soluble in both the aqueous and the oil phase, but is initially put in one of the phases. When

the surfactant is initially dissolved in the continuous phase, a phenomenon called cyclic

dimpling is caused by the diffusion flux. In the opposite case, when the surfactant is initially

dissolved in the drops, the diffusion brings about an osmotic swelling of the film. It is

interesting to note that in both cases the diffusion of surfactant across the interfaces leads to

stabilization of the liquid films and the emulsions.

Finally we consider the hydrodynamic theory of thin liquid film rupture. The stability

of the liquid films to a great extent is ensured by the property of the adsorbed surfactant to

damp the thermally excited fluctuation capillary waves representing peristaltic variations in

the film thickness [6]. In addition to the theory of stability of free foam and emulsion films,

we consider also the drainage and stability of wetting films, which find application in various

coating technologies [7].

II. THERMODYNAMICS OF ADSORPTION FROM SURFACTANT SOLUTIONS

The fluid dispersions (emulsions, foams) are stabilized by adsorption layers of

amphiphile molecules. These can be ionic and nonionic surfactants, lipids, proteins, etc. All

of them have the property to lower the value of the surface (or interfacial) tension, σ, in

accordance with the Gibbs adsorption equation [8-10],

d =- d lni ii

kT cσ Γ∑ (1)

where k is the Boltzmann constant, T is temperature, ci and Γi are respectively the bulk

concentration and the surface concentration (adsorption) of the i-th component in the solution.

The summation in Eq. (1) is carried out over all components. Usually an equimolecular

dividing surface with respect to the solvent is introduced; the adsorption of the solvent is set

zero by definition [8,9]. Then the summation is carried out over all other components. Γi

itself is an excess surface concentration with respect to the bulk. Γi is positive for surfactants,

which decrease σ in accordance with Eq.(1). On the opposite, Γi is negative for aqueous

solutions of electrolytes, whose ions are repelled from the surface by the electrostatic image

forces [9]; consequently, the addition of electrolytes increases the surface tension of water

4

[10]. When the interactions between the dissolved molecules/ions become important, ci in

Eq. (1) must be replaced with the respective activity.

A. Langmuir Adsorption Isotherm and its Generalizations

A relation between the bulk surfactant concentration, c, and the surfactant adsorption,

Γ, is provided by the Langmuir isotherm,

Γ Γ=+∞c

B c (2)

where Γ∞ is the maximum possible adsorption at c → ∞, and B is a constant, which can be

related to the energy of adsorption per molecule [10-12]:

0 0exps

BkT

∞ −Γ=

µ µ

δ (3)

Here δ is the thickness of the adsorption layer, 0sµ and µ0are the standard chemical potentials

of a surfactant molecule on the surface and in the bulk of solution; c and B are measured in

cm-3. In spite of the fact that Eq.(2) is originally derived for the special case of localized

adsorption of non-interacting molecules, it can be successfully applied to various surfactants,

including ionic ones [13-17]. The Langmuir isotherm can be generalized for multicomponent

adsorption [18].

On integrating Eq. (1), along with Eq. (2), one obtains the Szyszkowski [19] equation

relating σ and c,

π σ σs kT c B≡ − = +∞ ( )0 Γ ln / 1 (4)

Here σ0 is the surface tension of the pure solvent and πs is called the surface pressure. Finally,

by eliminating c between Eqs. (2) and (4) one derives the respective surface equation of state,

often called the Frumkin [20] equation:

ln 1s kTπ ∞∞

Γ= − Γ − Γ

(5)

Equations (2)-(5) take into account neither the interaction between the adsorbed

surfactant molecules, nor the electrostatic energy of the surfactant ion (in case of ionic

surfactant). Borwankar and Wasan [21] generalized the Langmuir isotherm, Eq. (2), to

account for the latter two effects:

exp -A exp /sZec B

kT∞ ∞

Γ Γ=

Γ − Γ Γ

ψ (6)

5

where Z is the valence of the surfactant ion (Z=0 for a nonionic surfactant), e is the

elementary charge, ψs is the surface electric potential (provided that the electric potential in

the bulk of solution is zero), A is a constant accounting for the non-electrostatic interactions

among the adsorbed amphiphiles. The combination of Eq. (6) with the general Gibbs equation

(1) and with the Gouy-Chapman theory of the electric double layer [3] yields the following

two-dimensional equation of state [21], generalizing Eq. (5):

( )2 3

2

8ln 1 cosh 1

2 2S

S

C kT eAkTe kT∞

∞ ∞

Γ Γ = − Γ − + + − Γ Γ

ε ψππ

(7)

where ε is the dielectric constant of the solution and the surface active ion, as well as the

added electrolyte, is assumed monovalent; C denotes the total electrolyte concentration

(surfactant + added salt) [21]. In the case of non-ionic surfactant one can use Eq. (7) after

setting formally ψs = 0. Note that ψs in Eqs. (6) and (7) depends on Γ: the connection between

surface charge and surface potential provided by the double layer theory [3],

2

8sinh

2sCkT

e

ekT

Γ =ε

π

ψ (8)

may be used.

There has been a discussion in the literature whether the non-ionic surfactants in

solution do bear some charge because of the adsorption of ions (say OH−) on the ethylene-

oxide chains. Recent precise electrophoretic measurements with air bubbles and oil droplets

in pure water or non-ionic surfactant solutions reveal that the observed negative surface

charge is an inherent property of the interface water-hydrophobic phase (air, oil), which can

be attributed to the adsorption of hydroxyl ions at the interface [22]. The non-ionic surfactant

molecules, themselves, are not charged, but they can reduce the surface charge by displacing

the adsorbed OH− from the interface; for details see Ref. [22] and the literature cited therein.

B. Van der Waals Type Adsorption Isotherms

Another type of adsorption isotherms are based on the two-dimensional van der Waals

equation of state [23],

( )2S a a kTa ∞

+ − =

βπ (9)

corresponding to the case of non-localized adsorption; here a = 1/Γ is the area per molecule,

a∞ = 1/Γ∞ is the excluded area per molecule and β accounts for the interactions between the

6

adsorbed molecules. The statistical theory [23,24] relates the parameters a∞ and β with the

potential of interaction, u(r), between two adsorbed surfactant molecules:

( ) 2

01 exp

2cr

c

u ra rdr r

kT∞

= − − ≈

∫

ππ (10)

( ) ( )1 expc cr r

u rkT rdr u r rdr

kT∞ ∞

= − − − ≈ −

∫ ∫β π π (11)

where rc is the intermolecular center-to-center distance at contact, i.e. u(r) → ∞ for r < rc .

Eq. (10) and (11) explicitly shows that a∞ and β account for the hard-core and long-range

interactions, respectively. Combining Eqs. (9) and (1) one obtains the relation between c and

a in the van der Waals approach:

2exp /a a c Ba a akT a a

∞ ∞

∞ ∞

− + = − −

β (12)

where the constant B is given by Eq. (3).

Note that the integral in Eq. (11) is convergent only when u(r) tends to zero faster than

r-2 for r → ∞, for example such is the case of dipole-dipole interaction. In the case of the

slowly decaying Coulombic interaction between adsorbed surfactant ions, generalized

versions of Eq. (9) and (12) can be derived

( )3

2 2

8cosh 1

2s

S

C kT ekTa a a e kT∞

= − + + − −

ε ψβππ

(13)

2exp /Sea a c Ba a akT a a kT

∞ ∞

∞ ∞

− + + = − −

ψβ (14)

Eqs. (13) and (14) are counterparts of Eqs. (7) and (6), respectively. For β = 0 Eq. (13)

reduces to the equation derived by Davies [25] and Hachisu [26]. In view of Eq. (.8) one finds

that the last term in Eq. (13) decreases, when the total electrolyte concentration C increases at

fixed area per molecule, a. This would lead to a decrease in πS, if a were really constant.

However, Eqs. (8) and (14) show that a decreases when C increases at fixed surfactant

concentration c, which leads to an increase of πS. The competition between these two opposite

trends determines the dependence of πS on the electrolyte concentration C. For sodium

dodecylsulfate the second trend (the decrease of a) is predominant and πS increases with the

increase of C at fixed c.

Expanding in series for a∞ /a << 1, from Eq. (9) one can derive

7

2

21 1Saa a

OkT kTa a a

∞∞

∞

= + − +

π β (15)

Hence, for a∞ /a << 1 the plot of πS a/(kT) vs. a∞ /a should be a straight line of intercept 1.

However, in some cases an intercept close to 1/2 can be found: see Ref.[27]. This can be

attributed to the formation of doublets of adsorbed molecules due to the component of their

dipole moments oriented laterally to the interface. This effect can be observed with

zwitterionic amphiphiles like some lipids [27]. A doublet represents a couple of adsorbed

molecules with antiparallel orientations of their lateral dipole moments.

In addition, it should be noted that various semi-empirical surface equations of state

can be used, see e.g. Ref. [28]. For example, Tajima et al. [29] proposed the following

equation for sodium dodecylsulfate:

( )( )1/ 2 0.436.03S

kTC a a kT

a∞+ + − =π (16)

with a∞ = 38.4 Å2. Similar dependence of πS on C can be deduced from Eqs. (8) and (13) for

high value of the surface potential ψs.

Finally, we give the general definition of Gibbs elasticity [8]:

E aaG = = −

∂σ∂

∂σ∂

ΓΓ

(17)

EG characterizes the increase of the surface tension with the quasistatic dilatation of the

adsorption monolayer.

For the reader's convenience some of the most frequently used adsorption isotherms

and surface equations of state (that of Henry, Langmuir, Freundlich, Volmer, Frumkin and

van der Waals) [35,49-51] are summarized in Table 1; the respective expressions for

/ cΓ∂ ∂ , and the Gibbs elasticity, EG, stemming from the various isotherms are also given;

ΓF, BF and m are characteristic parameters of the Freundlich adsorption isotherm.

8

Table 1. The most frequently used adsorption isotherms

• Adsorption isotherm Henry Γ Γ/ /∞ = c B Langmuir ( )Γ Γ/ /∞ = +c B c Freundlich ( )Γ Γ/ /F F

mc B= Volmer ( )[ ] ( )c B/ exp / /= − −∞ ∞Γ Γ Γ Γ Γ Γ Frumkin ( ) ( )c B kT/ exp / /= − −∞Γ Γ Γ Γ2β van der Waals ( )[ ] ( )c B kT/ exp / / /= − − −∞ ∞Γ Γ Γ Γ Γ Γ Γ2β • Equation of state Henry σ σ= −0 kT Γ Langmuir ( )σ σ= + −∞ ∞0 1kT Γ Γ Γln / Freundlich σ σ= −0 kT mΓ / Volmer ( )σ σ= − −∞ ∞0 kT Γ Γ Γ Γ/ Frumkin ( )σ σ β= + − +∞ ∞0

21kT Γ Γ Γ Γln / van der Waals ( )σ σ β= − − +∞ ∞0

2kT Γ Γ Γ Γ Γ/ • ∂ ∂Γ / c Henry Γ / c Langmuir ( )Γ∞ +B B c/ 2 Freundlich m cΓ / Volmer ( )Γ Γ Γ1 2− ∞/ / c Frumkin Γ Γ

Γ ΓΓ

c kT∞

∞ −−

2β

van der Waals

( )Γ Γ

Γ Γ

Γc kT

∞

∞ −−

2

22β

• EG = −∂σ ∂/ lnΓ Henry kT Γ Langmuir kT Γ Γ Γ/ ( / )1 − ∞ Freundlich kT mΓ / Volmer ( )kT Γ Γ Γ/ /1 2− ∞ Frumkin

kTkT

ΓΓ

Γ ΓΓ∞

∞ −−

2β

van der Waals

( )kT

kTΓ

Γ

Γ Γ

Γ∞

∞ −−

2

22β

9

III. KINETICS OF ADSORPTION FROM SURFACTANT SOLUTION

Not only the equilibrium surface tension, but also the kinetic properties of a surfactant

adsorption monolayer play an important role in various phenomena related to the stability of

foams and emulsions [5, 30], rising of bubbles and flotation [31]. Indeed, many processes are

accompanied by disturbances (expansion, compression) of the adsorption monolayer or by

formation of new surface of the solution. The surfactant solution has the property to damp the

disturbances by diffusion of surfactant from the bulk to the interface, or vise versa. The main

subject of the present Section III is the theory of adsorption and surface tension under such

dynamic conditions.

Different experimental techniques have been developed for the measurement of

dynamic surface tension. The most popular among them are the oscillating jet method [32],

the inclined plate method [33], the maximum bubble pressure (MBP) method [34-39], the

surface wave techniques [40-43], the oscillating bubble method [44-48], the drop weight

techniques [14, 49-51], and the free falling film method [52, 53]. Detailed reviews on the

experimental techniques can be found in the book by Dukhin et al. [31] and the paper by

Franses et al. [54].

As mentioned earlier, below we focus our attention on the kinetics of surfactant

adsorption. First we introduce the basic equations. Next we consider the two alternative cases

of surfactant adsorption: under diffusion and barrier control. Special attention is paid to the

adsorption of ionic surfactants, whose molecules are involved in long-range elecrtostatic

interactions. Finally, we consider the adsorption from micellar surfactant solutions, which is

accompanied by micelle diffusion, assembly or disintegration.

A. Basic equations The basic equation describing the transport of the i-th solute in solution is the mass-

balance equation (see Refs. 55-58)

( ) , 1, 2,..., ;i i ii i i i i

c q cc D c D R i Nt kT

+ ∇ ⋅ = ∇ ⋅ ∇ + ∇ + =

v∂ ψ∂

(18)

where ∇ is the spatial gradient operator, t is the time, v is the mean mass velocity, qi and Di

are the electric charge and the diffusion coefficient of the i-th solute and Ri is its rate of

production in chemical reactions; ψ is the potential of the electric field; N denotes the total

10

number of the solutes; the meaning of ci, k and T is the same as in Eq. (1). The term ∇ ⋅ ( )civ

in Eq. (18), called the convective term, accounts for the transport of component i by

hydrodynamic fluxes; this term is essential for some methods of dynamic surface tension

measurement accompanied by liquid flow, like the maximum bubble pressure method [14,35].

The terms with ∇ci and ∇ψ take into account the contributions of the diffusion and electric

current into the overall mass balance. Of course, the electric term is zero for the non-charged

species. Usually the liquids are treated as incompressible fluids, i.e. ∇⋅v = 0. Then the

convective term in Eq. (18) can be simplified:

∇ ⋅ = ⋅ ∇( )c ci iv v (19)

Note that the diffusivity, Di, and the kinematic viscosity of the liquid, ν, have the same

physical dimension (cm2/s). Their ratio Sc = Di /ν, called the Schmidt number, is usually

small, of the order of 10-3. The latter fact has the consequence that the diffusion boundary

layer is much thinner than the hydrodynamic boundary layer in dynamic problems [59].

A two-dimensional analogue of Eq. (18), expressing the surface mass balance, holds at

the interface:

( )∂∂

ψΓ

Γ ΓΓi

i s is

i is i i

i i stD D

qkT

r i N+ ∇ ⋅ = ∇ ⋅ ∇ + ∇

+ − ⋅ =II II II IIv n J , , ,..., ;1 2 (20)

Here ∇II is the surface gradient operator, Ns denotes the number of adsorbed components, Γi

is the adsorption of component i, vs is the surface velocity, Dis is the coefficient of surface

diffusion, and rk is the rate of production of component k by interfacial chemical reactions; n

is the running unit normal to the surface, Ji is the subsurface value of the bulk electrodiffusion

flux

J i i ii iD c

q ckT

≡ − ∇ + ∇

ψ ; (21)

the symbol ⟨Ji⟩ denotes the difference between the values of Ji on the upper and lower side of

the interface; the term with ⟨Ji⟩ in Eq. (20) accounts for the exchange of component i between

the adsorption layer and the two adjacent bulk phases.

Note that some of the components present at the interface may not be present in the

bulk. An example are the insoluble surfactants. Another example represent components which

appear and disappear only in specific interfacial chemical reactions (for example

conformation of the adsorbed protein molecules from ball to long loop structures [60]).

11

In general, Eq. (20) serves as a boundary condition for the diffusion problem; it

provides an expression for ∇ci at the surface. In addition, boundary conditions for ci can be

used. These can be adsorption isotherms, like Eqs. (2), (6) or (14), relating subsurface

concentration and adsorption. In the case of liquid-liquid interface one can use also a

boundary condition expressing the equilibrium partitioning of a given solute between the two

neighboring liquids [58]:

c K c i Ni i i( ) ( ) ( , ,..., )0 0 1 2+ = − = (22)

Here Ki is the surface partition coefficient of component i; ci(0+) and ci(0-) denote the values

of ci on the two sides of the interface. At equilibrium Eq. (22) connects the bulk

concentrations, whereas for dynamic processes (under diffusion control) Eq. (22) connects the

subsurface concentrations only [58].

In general, the surfactant adsorption is a consequence of two stages: the first stage is

the diffusion of surfactant from the bulk solution to the subsurface; the second stage is the

transfer of the surfactant molecules from the subsurface to the surface. The following

important cases of surfactant adsorption kinetics can be specified:

(i) Diffusion-controlled adsorption: the first stage (the surfactant diffusion) is much

slower than the second stage and, consequently, determines the rate of adsorption. This case,

which is very usual in practice, is considered theoretically in Subsection B below.

(ii) Barrier-controlled adsorption: the second stage is much slower than the diffusion

because of the presence of some kinetic barrier, which slows down the transfer of the

surfactant molecules from the subsurface to the adsorption monolayer. This barrier can be due

to steric hindrance, electrostatic repulsion or conformational changes accompanying the

adsorption of some molecules (usually polymers or proteins) - see Subsection C for details.

(iii) Electrodiffusion-controlled adsorption: this is the case of adsorption of ionic

surfactants. Since the extent of the electric double layer is much larger than the size of the

surfactant ions, the electrostatic interactions affect both the diffusion and the surfactant

transfer from the subsurface to the surface. This case needs a special theoretical treatment,

which is reviewed in Subsection D below.

(iv) Adsorption from micellar solution: in this case both surfactant monomers and

micelles are involved in the diffusion process. Thus the micelles serve as carriers of surfactant

from the bulk to the surface and vice versa. The reactions of micelle formation and decay

should be necessarily taken into account, see Subsection E for details.

12

B. Adsorption under Diffusion Control Let us consider the case of diffusion-controlled adsorption of electroneutral molecules

at an interface, which is subjected to uniform expansion or compression. The interfacial

expansion or compression is accompanied by hydrodynamic flux, which can be expressed by

the equation of van Voorst Vader et al. [61]:

v x x t= − & ( )α (23)

where the x-axis is directed perpendicular to the solution surface inwards, vx is the x-

component of the velocity v, A(t) denotes the interfacial area and & ( )α t is the rate of interfacial

dilatation. Then, in view of Eqs. (19), (21) and (23), one can transform Eqs. (18) and (20) to

read

∂∂

α∂∂

∂∂

∂∂

ct

xcx x

Dcx

x tk kk

k− =

> >& ( , )0 0 (24)

( 0, 0)k kk k

d cD x tdt x

∂α∂

Γ+ Γ = = >& (25)

To specify the initial conditions, let us assume that at the moment t = 0 the bulk

concentration is equal to the equilibrium one, ck e, , and that the initial adsorption of each

component, Γk,0, is known.

B.1. Kinetics of adsorption at quiescent surface

If the interfacial expansion (compression) happens only at the initial moment and then

the interface is quiescent (i.e. &α ≡ 0 for t > 0), the relaxation of the surfactant adsorption is

described by the expression of Ward and Tordai [62]:

( )[ ]Γ Γkk

k e k s

t

kD

c c t d k N= − − + =∫2 10

0πτ τ, , , ( ,2,..., ) (26)

which is a corollary of Eqs. (24) and (25); ck,s is the subsurface concentration and τ is a

variable of integration. Eq. (26) coupled with a given adsorption isotherm, Γk(ck,s), represents

a Volterra integral system of equations, which, in general, has no analytical solution. In the

case of one surfactant component a numerical procedure for solving the Eq. (26) with any

adsorption isotherm is presented in Refs. [63-65]; expansions in power series for some of the

most important isotherms are also available [66-68].

In the special case of single amphiphilic component and small deviations from

equilibrium, the boundary problem can be linearized and Eq. (26) yields [69]

13

Γ ΓΓ Γ

Γe

e

e

e

s e

s e r rr

c c

t t c t cc c

tt

tt

tD c

e

−−

=−−

=−−

=

≡

=

( )( )

( )( )

( )( )

exp ,0 0 0

1 2σ σσ σ

∂∂

erfc , (27)

where Γe is the equilibrium adsorption corresponding to bulk surfactant concentration ce, cs

is the subsurface concentration; tr is the characteristic relaxation time of adsorption, and

erfc(x) is error function, see e.g Ref. [70]. (Theoretical approaches to the case of large

deviations from equilibrium are described in section B.2 below.) The values of ∂Γ/∂c for

some of the most frequently used adsorption isotherms (that of Henry, Langmuir, Freundlich,

Volmer, Frumkin and van der Waals) are given in Table 1. Eq. (27) can be generalized for

micellar solutions, see Eq. (70) below.

The generalizations of Eqs. (26) and (27) to the case of surfactant soluble in two

adjacent liquid phases have been obtained by Miller [71]. It was demonstrated that the

distribution coefficient, Ki, see Eq. (22), can be determined from experimental data for the

dynamic interfacial tension. Since this coefficient is closely related to the HLB-value [72],

such experiments would be useful to characterize the application of surfactants as emulsifiers.

The problem for the diffusion controlled adsorption of a mixture of surfactants at

small deviation from equilibrium can be solved analytically by using the Laplace transform;

the resulting long expressions can be found in Ref. 73. The results are compared with

experimental data for different alkanoates at air-liquid interface [73]. Simple adsorption

isotherms for mixtures of surfactants are reported in Refs. [31, 44]; surface equations of state

applicable to mixtures of surfactant molecules of different size can be found in Refs. [27, 74].

When the initial deviation from equilibrium is not small, an exact general analytical

expression for σ(t) cannot be derived. However, explicit asymptotic expressions for long

times (t → ∞) and short times (t → 0) of adsorption can be obtained, even for surfactant

blends. The long time asymptotics of the subsurface concentration, ci,s,

( ) ( )c cD t

ot

i Ni e i s i e ii

, , , ,

/

( ) , ,...,− = −

+ =Γ Γ 0

1 21 1 2π , (28)

has been derived by Hansen [75], and further generalized in Refs. [31, 44, 73, 76, 77]. Here

Γi e, is the equilibrium adsorption corresponding to bulk concentration ci,e. It is to be noted

that many authors have reported Eq. (28) with a wrong factor 4 before Di (see Ref. 76);

however, the validity of Eq. (28) has been rigorously proved [78].

Using the Gibbs adsorption equation for a multicomponent solution (this is Eq. (1)

with ci ≡ ci,s), one can derive the long time asymptotics of the surface tension

14

( )σ σ π( ) ( ), , ,

,

/

tkT

c D to

tei e i e i

i e ii

N− =

−

+

=∑

Γ Γ Γ 01 2

1

1 (29)

where σe is the equilibrium surface tension. The assumption for independence on the various

surfactant species, implicitly used to derive Eqs. (28) and (29), might be violated for large

values of the adsorption. On the other hand, Eq. (29) is independent of the type of the

adsorption isotherm, cf. Table 1.

The short time asymptotics of the adsorption has the following form [31, 44, 73, 76,

77]

( ) ( )Γ Γi i i e iic c

D to t i N− = −

+ =, , ,

/( ,2,..., )0 0

1 241

π (30)

where ci,0 is the initial subsurface concentration corresponding to the initial adsorption Γi,0.

Then substituting Eq. (30) in the Gibbs adsorption equation one obtains the short time

asymptotics of the surface tension reads

( ) ( )σ σπ

( ) ( ) , ,

* /

t kT c cD t

o ti e ii

i

N− = − −

+

=∑0

40

1 2

1 (31)

where σ(0) is the surface tension at the initial moment, and the mean diffusion coefficients are

defined by expressions

( )D Dc

k Nk k ii

k c ci

N

i i

*,

ln,2,...,

,

=

===

∑ΓΓ0

1

2

0

1∂∂

(32)

In the special case of single surfactant and initially clean surface, Γ0 = 0, Eq. (29) and (31)

reduce to

σ σ π σ σπ

− ≈

− ≈

e

e

ee

kTc Dt

kT c DtΓ2 1 2 1 20 4/ /

, ( ) (33)

Eqs. (28)-(33) refer to the limiting cases t → ∞ and t → 0, however no analytical

expression is available for the intermediate times. Sometimes, empirical formulas are

employed to process experimental data for the intermediate times. Rosen et al. [79-81]

proposed the following empirical equation

σ σσ σ

0 1−

−= +

me

me

ntt* (34)

where σme is a "mesoequilibrium" surface tension of the solution (for which σ shows only

small changes with time), σ0 is the equilibrium surface tension of the pure solvent; t* and n

15

are adjustable parameters, which have been determined for a large amount of experimental

data [79-81].

A general (though approximate) theoretical approach to the calculation of Γ(t) and

σ(t) for any magnitude of the surface deformation is presented below, see Eqs. (41)-(42).

B.2. Kinetics of adsorption at expanding surface ( &α ≠ 0)

In many experimental techniques used for dynamic surface tension measurements

(such as the MBP method and the drop volume method [14, 76, 82]) the surface expands

gradually with time. In such a case the convective terms in Eqs. (24) and (25) cannot be

neglected. Nevertheless, it can be demonstrated that with the help of the new independent

variables,

( ) ( )( )

z xs t s d sA tA

t= = ≡∫( ) ,and whereθ τ τ2

0 0 , (35)

Eqs. (24) and (25) acquire the form of diffusion equations with constant coefficients:

∂∂θ

∂∂ θ

∂∂

cD

cz

d sd

Dcz

ii

i ii

i

z= =

=

2

20

,( )Γ

(36)

Eqs. (36) lead to a result, which is similar to the Ward-Tordai expression, Eq. (26):

( )[ ]sD

c c d i Nii

i e i s iΓ Γ= − − + =∫2 1 20

0πθ τ τ

θ

, , , ( , ,..., ) (37)

The solution of Eq. (37) for small surface deformations is obtained in Refs. [83, 84] and it is

applied there for the interpretation of experimental data at known expansion rate, s(t).

However, there is no analytical solution of Eq. (37) for the case of large surface

deformations. Nevertheless, analytical expressions for the long and short time asymptotics of

Eq. (37) can be derived. In the case of single component and initially clean surface, Γ(0) = 0,

the following short time asymptotics stems from Eq. (37):

( )Γ =+

=

→c Dt

ted

dt

42 1

1 2

0π αα α

/

, lim( & ) (38)

Substituting αd = 2/3 in Eq. (38), one obtains the Joos formula, which has been applied to

interpret experimental data from the MBP method [14, 76, 82].

A way to simplify the adsorption problem in the case of large surface deformations

has been proposed by Kralchevsky et al. [35]. In so far as one is interested in the time

dependence of the surface properties, Γ(t) and σ(t), one can use an appropriate model

16

expression, c*(x,t), for the surfactant concentration profile. Thus one avoids the time

consuming numerical integration of the partial differential equation of surfactant diffusion.

The function c*(x,t) must satisfy the following integral condition for equivalence between the

real and model concentration profiles [35]:

( ) ( )1 10 0

−

= −

≡

∞ ∞

∫ ∫c x t

cdx

c x tc

dx l te e

, * ,( ) (39)

Similarly to the von Karman boundary layer theory [85], the following model profile can be

used [35]:

[ ]c x t c t c c t x x

c x t c x

s e s

e

* ( , ) ( ) ( ) sin ;

* ( , )

= + −

≤

= ≥

πδ

δ

δ2

for

for (40)

where cs(t) is the subsurface concentration, see Fig. 2. Note the smooth matching of the sine

with the line c = ce = const at the point x = δ. The parameter δ is determined by substituting

Eq. (40) into Eq. (39). Finally, this approach leads to the following equation for Γ(t) [35]:

Γ Γ( ) ( ) [ ( ) ( )] / ( )t c l t c l s te e= + −0 0 (41)

where s(t) is a known function defined by Eq.(35) and l(t) satisfies the ordinary differential

equation [35]

dldt

t l tc tc

Dl t

s

e+ = −

−

& ( ) ( )

( )( )

α π2

1 12

(42)

Eqs. (41), (42) and the adsorption isotherm, cs= cs(Γ), (the latter can be any isotherm like

those in Table 1) form a set of three equations for determining the three unknown functions

Γ(t), l(t) and cs(t). The problem can be solved by numerical integration of Eq. (42). The initial

condition depends on the specific problem; for example, if only the surface, but not the

solution, is disturbed in the initial moment, i.e. c(x,0) = ce for x > 0, from Eq. (39) one obtains

l(0) = 0. Further, to determine the dynamic surface tension, σ(t), one uses the respective

surface equation of state, σ(Γ), cf. Table 1. This theoretical method is found to compare very

well with experimental data for dynamic surface tension of sodium dodecylsulfate solutions

measured by means of the MBP method [35].

The approach based on Eqs. (41)-(42) can be generalized to the case of micellar

solutions, see Eqs. (76)-(78).

17

Fig. 2. The model profile c* vs. x at a given moment t. c ts ( ) is the subsurface surfactant concentration; δ(t) is the point of smooth matching of the sine with constant, cf. Eq. (40).

C. Adsorption under barrier control

As mentioned above, the kinetics of adsorption is called barrier limited when the

stage of surfactant transfer from the subsurface to the surface is much slower than the

diffusion stage because of the existence of some kinetic barrier. This barrier can be due to

steric hindrance, electrostatic repulsion or conformational changes accompanying the

adsorption of the molecules. As the diffusion is the fast stage, the surfactant concentration in

the solution should be uniform and equal to that at equilibrium: c(x,t) = ce = const. In such a

case, the rate of increase of the adsorption, Γ(t), is solely determined by the "jumps" of the

surfactant molecules over the adsorption barrier separating the surface from the subsurface:

( ) ( )dd t

r c rsΓ

Γ Γ= −ad des, . (43)

Here rad and rdes are the rates of surfactant adsorption and desorption and , as usual, cs is the

surfactant subsurface concentration. (In a more general case of not too fast diffusion the

subsurface concentration, cs, can differ from the bulk concentration, ce.) Thus the problem is

reduced to the finding of appropriate expressions for rad(cs ,Γ) and rdes(Γ).

The first models of barrier controlled adsorption have been formulated by Doss [86]

and modified by Ross [87], both starting with the concept of Bond and Puls [88]. Further

developments have been achieved by Blair [89], Ward [90], Hansen and Wallace [91], and

Dervichian [92]. Baret et al. [93-95] analyzed the balance of the adsorption rad(cs,Γ) and

desorption rdes(Γ) rates and derived various analytical expressions related to known

equilibrium adsorption isotherms. In Table 2 we summarize the most frequently used

18

expressions for the rate of barrier limited adsorption, cf. Refs. [65,78,93-95]. In Table 2 Kad

and Kdes are the rate constants of adsorption and desorption, respectively, Ke = Kad/Kdes is the

equilibrium constant of adsorption, and the other notation is the same as in Table 1 above.

In fact, Table 2 contains model expressions; each of them can be substituted in the

right-hand side of Eq. (43) in order to calculate the rate of adsorption. In the simplest case of

very fast diffusion, when cs = ce.= const, the integration of Eq. (43) yields

( ) ( )Γ Γ Γ Γ( ) ( ) expt K te e= + − −0 des (Henry and Freundlich) (44)

( ) ( )des ad( ) (0) exp /e e et K K c t∞Γ = Γ + Γ − Γ − + Γ (Langmuir) (45)

Table 2. Kinetic models of reversible surfactant adsorption

• Total rate of surfactant adsorption Henry K c Ksad des− Γ Langmuir ( )K c Ksad des1 − −∞Γ Γ Γ/ Freundlich ( )K B c B KF s F

mad des/ − Γ

Frumkin K c

kT

KkT kT

sad

des

exp

exp

β

β β

Γ ΓΓ

ΓΓ

ΓΓ Γ

ΓΓ

∞

∞ ∞

∞

∞

−

−

−

−

2

2

1

2

• Equilibrium constant of adsorption Henry K Be = ∞Γ / Langmuir K Be = ∞Γ / Freundlich K Be F F= Γ / Frumkin K Be = ∞Γ /

Similar, but much longer, expression related to the Frumkin isotherm (Table 2) can be also

derived. In general, it turns out that in the case of barrier limited adsorption (Γe - Γ) decays

exponentially, in contrast with the case of diffusion limitation, in which (Γe - Γ) decays

proportionally to t-1/2: compare Eqs. (44)-(45) with Eqs. (28)-(29). The same is true for the

time dependence of the surface tension relaxation.

When the diffusion is not too fast, we deal with the more general case of mixed

diffusion-barrier control, which has been studied by Baret [95]. The solution of the diffusion

boundary problem in this case leads to the following equation

19

( )c cD

d td t

ds e

t= −

−∫

4 1 2

0πτ

τ/ Γ

, (46)

which together with Eq. (43) forms an integro-differential system. A numerical analysis of

this system has been performed in Ref. [96] by using the Henry and Langmuir models, as well

as in Ref. [65] on the basis of the Frumkin model, cf. Table 2. Quite different model was

proposed by Tsonopoulos et al. [97] and Yousef and McCoy [98], who treated the subsurface

region as an independent bulk phase with appropriate properties. Adamczyk [99,100] and

Filippov [78] generalized the model of mixed diffusion-barrier controlled adsorption and

derived some useful approximate expressions. Balbaert and Joos [101] investigated the

rheological aspect of the problem and demonstrated the applicability of the Boltzmann

superposition principle to the quantitative description of repeatedly compressing and

expanding adsorption layers of nonionic surfactants.

As an illustration below we give the solution of the mixed (diffusion-barrier) problem

employing the Henry model to characterize the barrier (see Table 2). In fact, the Henry model

is the only one which allows an exact analytical solution for Γ(t):

( ) ( ) ( )[ ]Γ Γ Γ( ) ( ) ( ) , ,t K c F b F be e= + − −0 0 2 1τ τ (47)

where τ = Kdest is dimensionless time, b1,2 = β ± (β2 – 1)1/2 are dimensionless parameters, with

β = Kad/(4DKdes)1/2 being a dimensionless diffusion-kinetic ratio coefficient; the function F is

defined as follows:

( ) ( ) ( )[ ]F bb

b bτβ

τ τ, exp=−

−1

2 11

22 erfc (48)

This solution implies that the characteristic time of the mixed, diffusion-barrier controlled,

adsorption is tch = 4D/ 2adK . In other words, the diffusion controls the kinetics of adsorption

when the time of the experiment is t >> tch, whereas the barrier controls the adsorption when

t << tch. In Fig. 3 the dependence of the dimensionless adsorption, Γ/Γe, on the

dimensionless time τ, calculated by means of Eq. (47), is shown for various values of the

parameter β. It is seen that the adsorption Γ is greater, when the diffusion is faster (at fixed

barrier), which is related to the fact that for larger diffusivity D the value of the parameter β

(denoted on the curves) is smaller.

20

D. Electro-diffusion control (adsorption of ionic surfactants)

Let us consider the process of adsorption of ionic surfactants at an quiescent (v = 0)

planar surface coinciding with the coordinate plane x = 0, see Fig. 4. In this case the equation

of electro-diffusion, Eq. (18), reduces to

( )∂∂

∂∂

∂∂

∂ ψ∂

ct x

Dcx

Dq ckT x

i Nii

ii

i i= +

= 1,2,..., (49)

Fig. 4. Sketch of adsorption monolayer of anionic surfactant. The charge of the surface headgroups is partially neutralized by adsorbed counterions.

Fig. 3. Dependence of the dimensionless adsorption, Γ/Γe, on the dimensionless time τ, for various values of the parameter β.

21

The concentration of the ionic species and the electric potential, ψ, are related by the Poisson

equation [1-4]:

εε ∂ ψ∂0

2

21x

q c x ti ii

N= −

=∑ ( , ) (50)

where ε and ε0 are the dielectric constants of the medium and vacuum, respectively. Eqs. (49)

and (50) form a nonlinear set of equations for determining the unknown functions ci(x,t) and

ψ(x,t).

The process of adsorption of ionic surfactants results in a continuous growing of the

surface charge density and the surface potential with time. In its own turn, the presence of

surface electric potential leads to the formation of an electric double layer inside the solution

[1-4]. The charged surface repels the new-coming surfactant molecules (see Fig. 4), which

results in a deceleration of the adsorption process. The respective boundary condition (for a

uniform adsorption monolayer) stems from Eqs. (20) and (21) [102-105]:

( )dd t

Dcx

q ckT x

i Nii

i i i

zs

Γ= +

=

=

∂∂

∂ ψ∂ 0

1,2,..., . (51)

An additional relation follows from the condition for integral electroneutrality of the

surfactant solution [2-4]:

εε ∂ ψ∂0

0 1xq

zi i

i

Ns

= == −∑ Γ (52)

The system formed by equations (49)-(52) has no analytical solution. Three types of

approaches to this problem can be found in the literature: (i) numerical methods [21, 103,

106, 107], (ii) approximate analytical expression derived by using an assumption for quasi-

equilibrium regime of surfactant adsorption [102-105], and (iii) exact asymptotic expressions

for the case of small deviations from equilibrium [108, 109].

The equilibrium distribution of the ionic species in the electric double layer (which is

the Boltzmann distribution) can be formally obtained by setting the left-hand side of Eq. (49)

equal to zero:

( )c x cqkT

i Ni e ii

, ,( ) exp ,2,...,= −

=∞ψ

1 (53)

Here ci,∞ denotes the surfactant concentration at large distance from the interface, where

ψ ≡ 0. The characteristic extent of the electric double layer is determined by the Debye

length, κ−1, with κ εε2 20= ∞∑ ( ) / ( ),q c kTi ii . Dukhin et al. [102-105] presented a quasi-

22

equilibrium model of the ionic surfactant adsorption. They assumed that the characteristic

time of the adsorption kinetics is much greater than the time of formation of the electric

double layer, tdl = 1/(κ2/D), as defined by Wagner [110]. Further, they simplified their task by

separating the diffusion from the electric problem:

for x > κ−1 : common diffusion in electroneutral solution

for x < κ−1 : kinetic barrier against surfactant adsorption due to the surface charge.

In other words, these authors reduce the electro-diffusion problem to a mixed barrier-

diffusion controlled problem (see the previous subsection). Such a simplification of the

problem is correct when the ionic strength of solution is high enough in order to have small

κ−1, i.e. the electric double layer is thin enough to be modeled as a kinetic barrier. Then one

can employ Eq. (46) to calculate the concentration at the "outer end" of the double layer (at x

= κ−1):

( ) ( )c t cD

d td t

di ii

tκ

πτ

τ−∞= −

−∫1

1 2

0

4, ,

/Γ

(54)

Finally an equation for barrier controlled adsorption is used [102-105]:

( ) ( )dd t

K c t K i Nii i i i s

ΓΓ= − =−

,ad ,, , ,...,κ 1 1 2des , (55)

cf. Eq. (43). The rate constant of adsorption, Ki,ad, depends on the surface electric potential.

Asymptotic solutions of Eqs. (54) and (55) are presented in Ref. [105].

When the diffusion time has comparable magnitude with the time of formation of the

electric double layer, the quasi-equilibrium model is not applicable. Lucassen et al. [111] and

Joss et al. [112] established that mixtures of anionic and cationic surfactants diffuse as

electroneutral combination in the case of small periodic fluctuations of the surface area;

consequently, this process is ruled by the simple diffusion equation. The electro-diffusion

problem was solved by Bonfillon et al. [113] for a similar case of small periodic surface

corrugations related to the capillary-wave methods of dynamic surface tension measurement.

Another problem, which has been solved analytically, is related to the relaxation of the

adsorption, Γ(t), and the surface tension, σ(t), of ionic surfactant solutions [108, 109]. In

general, the long time asymptotics reads

( )( )

( )( )

σ σσ σ π

e

e

e

e

rt t tt

−−

=−−

=0 0

Γ ΓΓ Γ

(t / tr » 1) (56)

23

where the subscript "e" denotes the equilibrium value of the respective quantity, and tr is the

relaxation time. In the case of a nonionic surfactant Eq. (56) can be deduced from Eq. (27)

with

tD cr

e=

1 2∂∂

Γ (nonionic surfactant) (57)

where D is the diffusivity of the surfactant molecules.

In the case of ionic surfactant solution, without additional electrolyte, Eq. (56) holds

again, but with the following specific expression for the relaxation time [108]:

tD cr s

s

s

=

+ − + −

∞

−1 12

12

32

2 2

*exp( ) exp∂

∂λΓ

ΦΦ

Φ

(ionic surfactant only) (58)

where Φs = |Zeψs /kT| is the equilibrium value of the dimensionless surface potential (the

surfactant is supposed to be a Z:Z electrolyte);

1 12

1 12 2

1

1D D D c cs e ss*;

exp( )≡ +

≡

=

−∞λ ακ ∂

∂ακ ∂

∂Γ Γ

ΦΦ

. (59)

Here cs is the subsurface concentration, D1 is the diffusivity of the counterions and α is the

apparent degree of dissociation of the surfactant adsorption monolayer accounting for the

fraction of the surfactant ionizable groups which are not neutralized by adsorbed counterions,

see Fig. 4. A formal transition to electroneutral surface, Φs → 0, α → 0, reduces Eq. (58) to

Eq. (57) with an effective diffusivity D* instead of D, cf. Eq. (59). The difference between D*

and D accounts for the electrolytic dissociation of the ionic surfactant.

In the case of ionic surfactant solution with added common electrolyte (like NaCl) Eq.

(56) holds again, but along with another, more complicated, expression for the relaxation time

[109]:

tD cr

eff s

=

∞

12

∂∂

Γ

Φ

(ionic surfactant + electrolyte) (60)

where Deff is defined as follows

( )( ) ( )1 12

11 1

212

322

2 2

DxD

x xD

xD

xeff

s ss

s ss s

s≡ +−

+ −−

+

× + − + −

−

**exp exp expΦ Φ

Φλ

1 12

1 1

1 2DxD D

xD

xc

c cs s

s**

;≡ + +−

≡

+surfactant

surfactant electrolyte (61)

24

where D2 is the diffusivity of the coions of the additional electrolyte. One can check that for

xs = 1, i.e. in the limiting case without additional electrolyte, Eq. (60) reduces to Eq. (58), as it

should be expected. On the other hand, for xs → 0, i.e. for a great amount of added

electrolyte, Eq. (60) reduces to the respective expression for nonionic surfactants, Eq. (57). In

other words, it turns out that in solutions of high ionic strength the ionic surfactants diffuse

and adsorb like nonionic surfactants.

E. Adsorption from Micellar Solutions At concentrations above the critical micellization concentration (CMC) the kinetics of

surfactant adsorption is strongly affected (accelerated) by the presence of micelles. In general,

the micelles exist in equilibrium with the surfactant monomers in the bulk of solution (Fig. 5).

A dilatation of the surfactant adsorption monolayer leads to a transfer of monomers from the

subsurface to the surface, which causes a transient decrease of the subsurface concentration of

monomers. The latter is compensated by disintegration of part of the micelles in the

subsurface layer (Fig. 5). These processes are accompanied by diffusion transport of

monomers and micelles driven by the concentration gradients.

Fig. 5. In a vicinity of an expanded adsorption monolayer the micelles release monomers in order to restore the equilibrium surfactant concentration on the surface and in the bulk. The concentration gradients give rise to diffusion of micelles and monomers.

25

Aniansson et al. [114-117] have theoretically described the micelles as polydisperse

species, whose growth or decay is determined by the reaction

A A AiK

Ki

i

i↔ +

+

−

−1 1 (i = 2,..., M) (62)

where Ki+ and Ki

− are, respectively, the rate constants of micelle growth and decay, and M is

the aggregation number of the largest micelles. It should be noted, that the experiments on

micellization show that the relaxation in a micellar solution is characterised by two relaxation

times [115, 117], which can be interpreted in the following way. The decay of a micelle is

considered as a consequence of fast and a slow process. The fast process is

A A A m nmk

mf → + >−1 1 , (63)

where kf is the respective rate constant. In other words, for m greater than a certain value n a

micelle can easily release a monomer and transform into micelle of aggregation number m-1

without destruction of the micellar aggregate. In contrast, the detachment of a monomer from

a micelle with the critical aggregation number, m = n, leads to a break of the micelle into

highly unstable fractions, which disintegrate to monomers. This is the slow process, which

can be characterises by the equation

A n Anks → 1 (64)

Here ks is the rate constant of the slow process, which can be c.a. 100 times smaller than kf

[115, 117].

Quantitatively, the presence of micelles influences the monomer concentration, c1(x,t),

through the rate of monomer production due to micelle decay, see the source term R1 in Eq.

(18). The most general theoretical description, related to Eqs. (18) and (62), is based on the

following set of equations:

( ) ( )∂∂

∂∂

ct

Dc

xK c K c K c c K ci i i i

i

M1

1

21

2 2 12

2 2 1 13

2= − − − −+ − +−

−

=∑ (65)

( ) ( )∂∂

∂∂

ct

Dc

xK c c K c K c c K ci

ii

i i i i i i i i= + − − −+−

−++

+−

+

2

2 1 1 1 1 1 1 (66)

where i = 2,3,…, M and K KM M++

+−= =1 1 0 . What concerns the boundary conditions for Eqs.

(65)-(66), we note that only surfactant monomers and none of the surfactant aggregates do

adsorb at the interface. This leads to the following boundary conditions for the diffusion

fluxes at the solution surface:

26

( )dd t

Dcx

cx

i MiΓ= = =1

1 0 2 3∂∂

∂∂

, , ,..., (67)

It is a difficult task to solve Eqs. (65)-(67). That is the reason why some physical

simplifications are usually made.

Most frequently used is the Lucassen [118] approach with monodisperse micelles (of

aggregation number m); this approach is related to the studies of bulk micellization kinetics

by Kresheck et al. [119], Muller [120], and Hoffmann et al. [121]. The model provides the

following expressions for the source terms in Eq. (18):

R mK c mK c R K c K cd m am

m d m am

1 1 1= − = − +, (68)

where Ka and Kd are the rate constants of micelle assembly and decay. Using the same model,

Miller [122] has solved numerically the respective diffusion equations for the monomers and

the monodisperse micelles. The computations are carried out for the Henry’s adsorption

isotherm (which is not suitable for typical surfactants above CMC) and for the Langmuir

adsorption isotherm. The numerical examples demonstrated that the adsorption relaxation in

the presence of micelles is faster than the relaxation at concentrations close to CMC.

For concentrations close to CMC and for concentrations much above CMC the rate

constant Kd can be related to the relaxation times of the slow and fast processes [123]:

K k c c K m nm

k c cd s m d f m≈ << ≈−

≥for for1 1; (69)

- cf. Eqs. (63) and (64) above; usually (m-n) is of the order of 5% of m.

In a number of works [124-126] the model of monodisperse micelles has been used to

determine the diffusivity of the micelles. Joos and van Hunsel [127] have used this model to

interpret experimental data for kinetics of adsorption obtained by the drop volume method.

These authors have demonstrated that the data can be fitted by means of a simpler expression

for R1 and Rm , which are linear with respect to the concentration, see also Refs. [77, 128,

129]. In fact, every reaction mechanism reduces to a reaction of "pseudo-first" order for small

deviations from equilibrium, see e.g. Ref. (130), Section 5.

The general problem based on Eqs. (65)-(67) has been solved by several authors,

[130-134], for the special case of small deviations from equilibrium, and some analytical

expressions have been derived. For example, the following expression for the relaxation of

the surface tension of a micellar solution has been obtained [130]:

( ) ( )[ ] ( )σ σσ σ

τ τ τ( )( )

( )( )

, , expt t

g gE g E ge

e

e

e

−−

=−−

=−

− −0 0

1

1 21 2

Γ ΓΓ Γ

(70)

27

where E(g,τ) = gexp(g2τ)erfc(g√τ), τ = t/td, τ = t td/ , g1,2 = [1 ± (1 + 4β)1/2]/2, ,β = td/tm, and

[ ]t K m c cm d m e e= +−

( / ), ,1 21

1, and ( )t c Dd e= ∂ ∂Γ / /1

21 (71)

are the characteristic relaxation times of micellization and diffusion, see Ref. [119]; as usual

the subscript "e" denotes equilibrium values and m is the micelle aggregation number; Kd is a

rate constant of micelle decay. Equation (71) is a generalization of Eq. (27) for the case of

micellar solutions; indeed, the absence of micelles is equivalent to set β = 0; then g1 = 1,

g2 = 0, and Eq. (70) reduces to Eq. (27).

The validity of Eq. (70) is restricted to the case of small deviations from equilibrium

and quiescent surface ( &α = 0 ). A quantitative theory, which is not subjected to the latter two

restrictions, can be developed by generalizing the approach of the model concentration

profiles from Ref. [35]; this is done in Ref. [123]. In view of Eq. (68), the generalization of

Eqs. (24)-(25) to micellar solutions reads:

∂∂

α∂∂

∂∂

ct

xcx

Dc

xm K c m K c x td m a

m1 11

21

2 1 0 0− = + − > >& , ( , ) (72)

∂∂

α∂∂

∂∂

ct

xcx

Dcx

K c K c x tm mm

md m a

m− = − + > >& ( , )2

2 1 0 0 (73)

dd t

Dcx

x tΓΓ+ = = >& ( , )α

∂∂1

1 0 0 (74)

In so far as one is interested in the time dependence of the surface properties, Γ(t) and σ(t),

one can use appropriate model expressions for the monomer and micelle concentration

profiles, c1∗ (x,t) and cm

∗ (x,t). Thus the time consuming numerical integration of the partial

differential equations (72)-(73) can be avoided. The functions c1∗ (x,t) and cm

∗ (x,t) must satisfy

the following integral condition for equivalence between the real and model concentration

profiles, analogous to Eq. (39) [123]:

( ) ( )1 11

10

1

101−

= −

≡∗∞ ∞

∫ ∫c x t

cdx

c x tc

dx l te e

, ,( )

, ,; ( ) ( )1 1

0 0

−

= −

≡∗∞ ∞

∫ ∫c x t

cdx

c x tc

dx l tm

m e

m

m em

, ,( )

, ,

(75a)

The model function c1∗ (x,t) is defined [123] by an expression, which is completely analogous

to Eq. (40). The model function cm∗ (x,t) is defined by the expression

28

[ ]c x t c c t c c t x x

c x t c x

m m e m s m e m sm

m m e

∗

∗

= + − −

≤

= = ≥

( , ) [ ( )] ( ) cos ;

( , ) ;

, , , ,

,

12

12

πδ

δ

δ

for

const for

(75b)

where cm,s(t) is the micelle subsurface concentration. The cosine in Eq. (75b) guarantees the

fulfillment of the boundary condition, Eq. (67), for i = m. The model profiles c1∗ (x,t) and

cm∗ (x,t) are visualized in Fig. 6.

Fig. 6. Model concentration profiles, c1∗ (x,t) and cm

∗ (x,t), of the free monomers and micelles at a given moment t. The interface is located at x = 0; δ1(t) and δm(t) are the points of smooth matching of the sine (Eq. 40) and cosine (Eq. 75b) with the respective constant concentrations in the bulk. Note the smooth matching of the cosine with the line cm = cm,e = const at the point x = δm.

The positions of the matching points, δ1(t) and δm(t), are determined from Eq. (75a). With the

help of these model profiles the initial set of equations, Eqs. (72)-(74), reduces to the

following relatively simple set of algebraic and ordinary differential equations [123]:

( ) ( ) ( ) ( )Γ Γt c l t c l t s te m e m= + +1 1 0, , / ( ) (76)

( ) ( )d ld t

lD

lc tc

mc

Q ts

e e

11

1

1

1

1

2

1

22

1+ =−

−

−α

π. ,

, ,

( ) (77)

( )d ld t

lc

Q tmm

m e+ =α

. 1

, (78)

( )[ ] ( ) ( )d c

d tD

c lc c t K c t K c tm s m

m e mm e m s d m s a s

m,

,, , , ,= − − +

π 2

2 23

18 (79)

29

s(t) is the same as in Eq. (35). Equations (76)-(79), together with the adsorption isotherm,

c1,s(Γ) (the latter can be any isotherm like those in Table 1), form a set of 5 equations for

determining the 5 unknown functions, Γ(t), l1(t), lm(t), c1,s(t) and cm,s(t). The term Q(t) in Eqs.

(77) and (78) is defined as follows:

QK c l

c cG

cc

K c la em

e sm

s

ed m e m=

− −

−

+22

11

1

1 1

1

1

,

, ,

,

,,( )( )π

;

( ) ( )[ ]{ }G dmmξ ξ ξ ω ω

π≡ − + −∫

0

21 1

/sin

The system of Eqs. (76)-(79) can be solved numerically, and for the sake of convenience the

special function Gm(ξ) can be tabulated in the computer memory; a detailed description of the

numerical procedure can be found in Ref. [123].

The above theory was applied for interpretation of dynamic surface tension data

obtained with solutions of sodium dodecyl sulfate (SDS) by means of the MBP method. The

empirical adsorption isotherm, c1,s(Γ), of SDS due to Tajima [29] was used with a value

m ≈ 77 of the mean aggregation number of the micelles. The best numerical fits of the data

are shown in Fig. 7. Curves "a" and "b" correspond to surfactant concentrations below CMC;

that is the reason why the respective data are processed by means of Eqs. (41)-(42).

Fig. 7. Dynamic surface tension, σ, vs. the surface age, tc, of submicellar (a,b) and micellar (c,d) solutions of SDS in the presence of 0.128 M NaCl measured by the maximum bubble pressure method at surfactant concentrations: (a) 0.2 mM, (b) 0.4 mM, (c) 1.5 mM, and (d) 2.0 mM. The solid and empty symbols correspond to different runs (after Ref. 84).

30

The calculated from the curves diffusion coefficient of the SDS monomers is

D1 ≈ 5 × 10−6 cm2/s, which is close to the value determined by other authors [135]. This

value of D1 has been further used to fit the data for concentrations above CMC by means of

Eqs. (76)-(79), see curves "c" and "d" in Fig. 7. Thus from the latter two curves one

determines Kd ≈ 70 s-1 for the rate constant of micelle decay, which in view of Eq. (69) yields

kf ≈ 1400 s-1 for the characteristic time of the fast relaxation process of micellization.

F. Rheology of Adsorption Monolayers All kinetic considerations in the previous sections have been based on the mass

balance at the surface of a surfactant solution, Eq. (18). In the present subsection we proceed

with the interfacial momentum balance and its physical implications. We consider below the

theoretical description of the effects of elasticity and viscosity of surfactant adsorption

monolayers, which are known as the effects of Marangoni [136] and Boussinesq [137]. These

effects are important for many practical processes, such as dynamics and stability of

emulsions and foams, spraying and atomization, flotation of ores, coating processes, etc. [28,

58, 138, 139]. The surface balance of the linear momentum can be expressed in the form [57,

58, 140, 142]:

[ ]n T T T n⋅ − = ∇ ⋅ +( ) ( ) $1 2II s Π (80)

where T(1) and T(2) are the bulk stress tensors of the two neighboring phases, n is a running

unit normal to the interface directed from phase 1 to phase 2; Ts is the interfacial stress

tensor; Π is the disjoining pressure, which is operative only in thin liquid films; $n is the

normal to the reference surface of the film, see Ref. [143].

The basis of the interfacial rheology is the constitutive relation for the interfacial

stress tensor Ts , which can be written as a sum of deviatoric and isotropic parts [137, 140,

144, 145]:

T I D I vs dil s sh s s s= + + − ∇ ⋅

( )σ τ η2 1

2 II (81)

where σ is the interfacial tension, τdil is dilatational interfacial viscous stress (see Eqs. 85 and

86 below), III is the surface unit tensor,ηsh is the interfacial shear viscosity; Ds is the surface

rate-of-strain tensor, defined as

( ) ( )[ ]D v I I vs s sT= ∇ ⋅ + ⋅ ∇

12 II II II II (82)

31

Eq. (81) is analogous to the constitutive relation of a bulk viscous fluid, see e.g. [146]; in

particular, σ and ηsh are counterparts of the bulk isotropic pressure, P, and viscosity, η.

Historically, the necessity of accounting for special surface viscous effects, appeared

after the experiments by Lebedev [147] and Silvey [148], which showed a contradiction with

the theory of gravitational settling of emulsion droplets by Rybczynski [149] and Hadamar

[150]. This contradiction was resolved by Boussinesq [137], who took into account the excess

visco-elastic properties of the interface due to the presence of surfactant adsorption

monolayer. Further, experimental methods for measurements of the surface viscosity have

been developed.

F.1. Methods for measuring shear surface viscosity:

(i) The deep-channel surface viscometer is a frequently used experimental method for

measuring interfacial shear viscosity owing to its sensitivity (ηsh ≥ 10-4 sp) and relatively

simple analytical theory. The main drawback of this technique is the necessity of placing a

small tracer particle within the interfacial flow field for tracking the central surface velocity.

This may be particularly cumbersome with heavy oil systems, for which the particle may

require several hours or more to execute a complete revolution, as well as with liquid-liquid

systems, for which the placement of the particle at the interface may be difficult. For more

details see refs. 58, 151-156.

(ii) The disk viscometers are generally less sensitive devices (compared to the channel

method) for measuring interfacial shear viscosity (typically ηsh ≥ 10-2 sp) and exhibit a more

complex flow field. The primary advantage of the disk techniques is their ability to directly

measure the torsion stress. The exact placement of the disk at the interface and avoidance of

contact angle anomalies are problematic issues with this techniques. The disk viscometers

appear to be most useful for the measurements of ηsh of highly viscous interfacial adsorption

layers [157-162].

(iii) The knife-edge viscometers are similar to the disk viscometers in most respects. A

notable exception is the rotating wall knife-edge viscometer, which entails the measurement

of the velocity of a tracer particle within the fluid interface and is quite sensitive to interfacial

shear viscosity measurements, ηsh ≥ 10-5 sp. This device presently is the most promising

surface viscometer. For more details see refs. 58, 153, 154, 158, 164-170.

(iv) A new method for the measurement of low surface shear viscosities is described

in Refs. [171, 172]. This method is based on recording the sliding of a small spherical particle

32

down an inclined capillary meniscus formed in the vicinity of a vertical plate. The theory of

the method employs accurate expressions for the capillary force exerted on the floating

particle [173, 174], which is counterbalanced by the hydrodynamic drag force [175]. The

experiment [172] gives values of the drag coefficient which are in good quantitative

agreement with the hydrodynamic theory for pure liquids [175]. The addition of surfactant

strongly increases the drag coefficient. The latter effect is used to measure the surface

viscosity (ηsh ≥ 10-5 sp) of low molecular surfactants, such as SDS or Brij, whose surface

viscosity is not accessible to the accuracy of the most of the other methods [173].

F.2. Dilatational surface viscosity

Interfacial dilatational stress is measured in processes of isotropic expansion

(compression) of an interface. Such processes are realized in the maximum bubble pressure

method [176-180], the oscillating bubble method [181-183], the pulsed drop method [184],

and the drop expanding method [39, 83, 84, 185-187]. Because of the simple spherical

symmetry Eq. (81), together with the projection of Eq. (80) along n, yields

( ) ( ) ( ) [ ]T I Is dilt tR t

P P= + = −[ ] ( ) ( )σ τ II II21 2 (83)

where P(1) and P(2) are the pressures inside and outside the droplet or bubble, and R is its

radius. (At static conditions τdil = 0 and Eq. (83) reduces to the known Laplace equation of

capillarity.) For small deformations σ depends linearly on the strain, whereas τdil depends

linearly on the rate-of-strain; therefore, the contributions of σ and τdil in Eq. (83) can be

separated. In other words, σ and τdil account for the elastic and viscous effects accompanying

the dilatation (compression) of the interface. In view of Eq. (17), for small deformations one

can write:

σ σ ε ε δ= + ≡e GE a

a, (84)

where EG is the Gibbs elasticity, a is the area per surfactant molecule, and ε is the relative

dilatation of the adsorption monolayer. There are two alternative expressions (constitutive

relations) for the dilatational stress:

τ η α ααdil d AdAdt

= =, & , &1 (Refs. 137, 140) (85)

where A is the total interfacial area, and

τ η ε εεdil d adadt

= =, & , &1 (Refs. 83, 84) (86)

33

where ηd,α and ηd,ε are coefficients of surface dilational viscosity. As illustrated in Fig. 8, for

soluble surfactants &ε is always smaller than &α because of the diffusion supply of surfactant

molecules from the bulk of solution. In the limiting case of slow dilatation &ε = 0, even if

&α ≠ 0, because the diffusion succeeds to saturate the surface with surfactant and there is no

actual change in the density of the adsorption monolayer. In the other limit of fast dilatation

the diffusive surfactant supply is negligible and &α ≈ &ε . For insoluble surfactants &α ≡ &ε .

From a physical viewpoint the definition (86) of surface dilational viscosity seems to be more

reasonable because &ε accounts for the real variation of the distance between the adsorbed

molecules in the process of surface dilatation or compression [83, 84], cf. Fig. 8.

Fig. 8. The relative expansion of the area per molecule, δa/a, is smaller than the respective expansion of the total area, δA/A, because of the diffusion supply of surfactant molecules from the bulk of solution.

The analogue of ηd,ε (or ηd,α) in the three-dimensional (bulk) rheology is the so called "second

viscosity", which is responsible for the decay of the intensity of sound in viscous fluids [146].

Using Eqs. (84)-(86) one can obtain from Eq. (83) two alternative forms for the

variation of the isotropic surface stress:

δ ε η α δ ε η εα εT E T Es G d s G d= + = +, ,& , &or (87)

In fact, δTs can be directly determined by measuring the pressure inside the expanding droplet

(bubble) and its radius; indeed, in view of Eq. (83) one can write

[ ]δT R t P t R P P P Ps = − ≡ −12

0 0 1 2( ) ( ) ( ) ( ) ; ( ) ( )∆ ∆ ∆ (88)

In addition, &α can be also experimentally determined from the recorded expansion of the

droplet (bubble). However, for soluble surfactants it is very difficult to directly measure ε(t)

34

or &ε (t). Instead, one can theoretically calculate ε(t) from the equations of diffusion;

combining Eqs. (27) and (37) one obtains [83]

εθ θ

θθ=

−

−

≡∫ ∫exp

( ) ( ), ( ) ( ' ) '

( ) t tt

t tt

dsd

d t t s t dtA

r

t tA

rA

tA

0

2

0

erfc (89)

where tr is the diffusion relaxation time, see Eqs. (57)-(60), and s(t) (which is supposed to be

known) is defined by Eq. (35).

It should be noted that for low molecular surfactants the viscous term in Eq. (87) is

usually negligible. The relaxation of δTs in this case is entirely determined by the diffusion

relaxation of ε(t), as given by Eq. (89), rather than to a real surface viscous relaxation related

to ηd,ε (or ηd,α). Therefore, from such a relaxation process one can determine the diffusion

relaxation time tr, rather than the true surface dilatational viscosity, ηd,α or ηd,ε.

Eqs. (87)-(89) can be used to interpret data from expansion-relaxation experiments,

see e.g. Refs. [39, 83, 84]. Fitting the experimental data for the interfacial dilatation one can

in principle determine the Gibbs elasticity, EG , the diffusion relaxation time, tr, and the

dilatational surface viscosity, ηd,ε (or ηd,α). . The latter is accessible to the accuracy of the

aforementioned experimental techniques for high molecular surfactants and proteins;

sometimes, ηd,ε (or ηd,α) can be determined also for low molecular anionic surfactants, but in

the presence of multivalent counterions (like Ca2+ or Al3+), which link neighboring surfactant

headgroups.

A typical experiment can be constituted of the following three stages: (i) an initial

formation of a saturated surfactant adsorption layer at the interface keeping the drop area

constant for several hours, (ii) an expansion of the adsorption layer increasing the drop area

for several seconds, and (iii) a relaxation of the expanded adsorption layer under a constant

drop area. To illustrate the result of such an experiment, in Fig. 9 we present data from Ref.

[83, 84] for bovine serum albumin (BSA) adsorption layers at decane-water interface. The

influence of pH and the ionic strength on the rheological properties of BSA is investigated. In

Fig. 9 the values of EG , tr and ηd (in this case ηd = ηd,ε ≈ ηd,α) are normalized by their values

at pH = 5. The experimental data show that both the surface elasticity, EG , and relaxation

time, tr, increase with increase of pH. The interfacial dilatational viscosity, ηd , exhibits a

maximum at pH = 6. A similar peak of the interfacial shear viscosity of BSA at pH = 6 has

been observed by Graham and Phillips [188] at petroleum ether-water interface. The results in

Fig. 9 demonstrate a marked influence of the ionic strength on the rheological parameters.

35

pH

rela

tive

surfa

ce e

last

icity

E G/E

G(p

H=5

)

0.95

1.00

1.05

1.10

1.15

1.20

5 6 7 8 9

pI=4.9

I=0.2 M

I=0.07 Ma

pH

rela

tive

rela

xatio

n tim

et D

/t D(p

H=5

)

0

2

4

6

8

10

12

5 6 7 8 9

pI=4.9

I=0.07 M

I=0.2 M

c

pH

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

5 6 7 8 9

pI=4.9

I=0.2 M

I=0.07 M

b

Fig. 9. Interfacial elasticity, EG , diffusion relaxation time tr, and interfacial dilatation viscosity,ηd , vs. pH of solutions of 0.0125 wt% BSA; the other phase is decane. pH is maintained by phosphate buffer;the ionic strength, I, is adjusted by NaCl. The droplet expansion method is applied (after Ref. 84).

36

One encounters the following difficulties in the interpretation of the data from the

experiments with interfacial dilatation. As discussed in Ref. [58], the shear viscosity , ηsh,

does not influence the total stress, δTs, only for interfacial flow of perfect spherical

symmetry. If the latter requirement is not fulfilled by a given experimental technique, its

output data will be influenced by a mixture of dissipative effects (not only ηd, but also ηsh

and tr). The apparent interfacial viscosity thus determined is not a real interfacial property in

so far as it depends on the specific method of measurement. For example, the apparent

interfacial viscosity measured by the capillary wave methods [189-196] depends on the

frequency; the apparent interfacial viscosity measured by the Langmuir trough method [197,

198] is a sum of the dilatational and shear viscosities (ηd + ηsh); for the methods employing

non-spherical droplet deformation, like the spinning drop method [199-201], the apparent

surface viscosity is a complex function of the dilatational and shear interfacial viscosities.

As an illustration let us consider the case of small droplet oscillating with a given

frequency ω. By substituting the solution of the respective diffusion problem into Eq. (87) we

derive the following relation

( ) ( )[ ] ( )δ ωη ωη δα δα αT E q i qE

qq

dts G d d G

t= − + +

−

+≡ ∫

11 2

0

; & (90)

where i = −1 is the imaginary unit, and q-1 = 1 + [2ωtr]1/2 is a dimensionless parameter

related to the characteristic diffusion time, tr, cf. Eqs. (57)-(60), and the frequency of the

droplet oscillations, ω. For high frequencies q is very small (q → 0) and δTs/δa = EG + iωηd;

i.e. the real and the imaginary parts of the derivative δTs/δa give the true Gibbs elasticity,

EG , and dilatational viscosity ηd. If the frequency is not so large, then the real and the

imaginary parts of δTs/δa are combinations of EG, tr ,ηd, and ω, see Eq. (90), which should

not be termed "Gibbs elasticity" or "dilatational viscosity".

IV. FLEXURAL PROPERTIES OF SURFACTANT ADSORPTION MONOLAYERS

In the previous section we considered the response of a surfactant adsorption

monolayer to dilatation and shear deformations. In addition, in the present section we will

focus our attention on the properties of interfaces subjected to flexural deformations, i.e.

37

bending and torsion, see Fig. 10. Indeed, the interactions between the head-groups and the

tails of adsorbed surfactant molecules lead to a non-zero work of bending or torsion.

Fig. 10. Modes of deformation of a surface element: dilatation, shear, bending and torsion.

It should be noted that usually the energy of interfacial bending is much smaller than

the energy of dilatation. That is the reason why the interfacial flexural properties are

important for phase boundaries of low surface tension and/or high curvature, such as

surfactant micelles, microemulsions, emulsions, as well as lipid monolayers, bilayers and

biomembranes.

Below we first introduce the most general mechanical description of the surface

moments (torques) exerted on the boundary between two fluid phases. Then we consider the

thermodynamics of a curved interface (membrane) in terms of the work of flexural

deformation. Next we specify the bending rheology by means of the model of Helfrich [202].

Finally we review the available expressions for the contributions of the electrostatic, steric

and van der Waals interactions to the interfacial bending moment and curvature elastic

moduli. These expressions relate the interfacial flexural properties to the properties of the

adsorbed surfactant molecules.

A. Mechanical Description of a Curved Interface

Two approaches, mechanical and thermodynamical, exist for the theoretical

description of general curved interfaces and membranes. The first approach originates from

the classical theory of shells and plates, reviewed in Refs. (202, 204). The surface is regarded

38

as a two-dimensional continuum whose deformation is described in terms of the rate-of-strain

tensor and the tensor of curvature. In addition, the forces and the force moments acting in the

interface are expressed by the tensors of the interfacial stresses, σ, and moments (torques), M.

Figure 11 illustrates the physical meaning of the components of the latter two tensors. Usually

they are expressed in the form

( ) ; n M= + =a a a n M a aαβ µ αβσ σα β µ α βσ (91)

where Greek indices take values 1,2 (summation over repeated indices is supposed); a1, a2 are

surface covariant base vectors and n is the running unit normal to the surface; σ µ(n) (µ = 1,2)

are called transversal shear stresses.

Fig. 11. Components of the tensors (a) of the surface stresses, σ, and (b) of the surface moments (torques), M.

Tensor analysis and differential geometry can be applied in a straightforward manner

to derive the interfacial balances of linear and angular momentum. The general formalism can

be found in Refs. [203, 205]. Here we will only mention the results for the case of quasistatic

processes (negligible angular acceleration), when σαβ and Mαβ are symmetric surface tensors

which are both diagonal in the basis of principal curvatures. In these conditions the normal

projection of the angular momentum balance is identically satisfied [205], and the tangential

projection reads [203-206]:

( ),

n M= −α αβσ β (92)

(the comma denotes covariant derivative). Obviously, the transversal shear resultants are

connected with the presence of moments, which are not constant along the surface. The

39

normal and tangential projections of the linear momentum balance have the following form

[203-206]:

,b M P PII I− = −αβ αβσαβ αβ (93)

0, ,b M+ =αβ βγασ β β γ (94)

Here bαβ is the curvature tensor, and PII−PI is the pressure difference across the interface.

Eq.(93) is the generalized Laplace equation, valid in the presence of moments. On spherical

fluid interfaces Mαβ,αβ = 0, due to the symmetry; the curvature tensor is isotropic,

bαβ = Haαβ (aαβ are components of the surface metric tensor, and H is the mean curvature,

H = ½ aµν bµν). Then Eq.(93) reduces to

2H PII PIσ = − (95)

with σ being the scalar isotropic tension, σ = ½ aµν σµν.

For a complete mechanical description of the surface one needs to specify expressions

for the stresses and moments. This is usually done by postulating some constitutive relations

between stress and strain, which pertain to a particular model for the rheological behavior of

the interface. An example is the Scriven's constitutive relation [140] for σ, see Eq. (81) above.

In Section C below we discuss a constitutive relation for the tensor of the surface moments,

M. Before that we consider the thermodynamics of the curved interfaces.

B. Thermodynamics of Adsorption Monolayers and Membranes

The general thermodynamics of systems containing phase boundaries (including three-

phase contact lines) is due to Gibbs [8]. The bulk phases in the idealized system are

considered to be homogeneous up to sharp mathematical dividing surfaces. Excesses of all

extensive properties (such as the internal energy, U, the free energy, F, the entropy, S, the

number of molecules of the i-th component, Ni, etc.) are ascribed to the dividing surfaces. The

latter are considered as separate surface phases with their own thermodynamic fundamental

equations. The work for mechanical deformation is also taken into account. As far as the

surface rate-of-strain tensor and the curvature tensor are two-dimensional, each of them has

two independent scalar invariants. Therefore, one may distinguish exactly four independent

40

modes of surface deformation (Fig. 10): dilatation, shear, bending and torsion provide

separate contributions to the mechanical work per unit area, δws [207-208]:

sw B H D= + + + Θδ γ δ α ζδβ δ δ (96)

This equation is valid locally, at each point of the surface. δα is the relative dilatation of a

surface element dA, δα = δ(dA)/dA ; δβ is connected with the deviatoric part of the surface

rate-of-strain tensor [207] and characterizes the shear deformation. H and D are the mean and

deviatoric curvatures:

H c c D c c= + = −12 1 2

12 1 2( ) ( ) ; (97)

with c1, c2 being the two principal curvatures at a certain point of the surface. The coefficients

γ and ζ have the meaning of thermodynamic interfacial tension and shearing tension,

respectively. B and Θ represent the bending and torsion moments [207].

The basic idea of the local thermodynamic description, which is due to Gibbs [8], is to

apply the fundamental equation of a uniform surface phase locally, i.e. for each elementary

portion, dA, of the curved interface:

( ) ( ) ( ) ( )s s s sdU T dS dN w dAi ii= + +∑δ δ µ δ δ (98)

Here T and µi are temperature and chemical potentials; dUs, dSs and dNis denote the excess

surface internal energy, entropy and number of molecules of the i-th component, belonging to

the elementary parcel dA; the symbol "δ" denotes infinitesimal variation due to the occurrence

of a thermodynamic process in the system. Then one obtains [143]

( ) ( ) ( )s s s sdU u dA u u dA= = +δ δ δ δα (99)

where us is the surface density of Us. Similarly, one can derive

( ) ( ) ; ( ) ( )s s s sdS s s dA dN dAi i i= + = Γ + Γδ δ δα δ δ δα (100)

with ss and Γi being the surface densities of Ss and Nis, respectively. The substitution of Eqs.

(96), (99) and (100) into Eq. (98), after some transformations, yields [207, 208]

( )s s ss T B H Di ii= − − Γ + − + + + Θ∑δω δ δµ γ ω δα ζδβ δ δ (101)

41

where ωS ≡ uS – TsS - i ii

Γ∑µ is the density of the surface excess grand thermodynamic

potential.

Let us consider now fluid interfaces composed of chemical components, which are

soluble in (and equilibrated with) the adjacent bulk phases. If a new piece of area is created at

constant T, µi, β, H and D, this does not correspond to any change of the physical state of the

interface. In such a case (∂ωS/∂α) = 0 and from Eq.(101) one realizes that γ = ωs. Then Eq.

(101) reduces to a generalized form of the Gibbs adsorption equation, cf. Eq. (1). It is now

evident that the bending and torsion moments, B and Θ, are connected with the curvature

dependence of the interfacial tension.

The mechanical and thermodynamical approaches are two complementary routes for

description of interfaces and membranes of arbitrary shape. It is very important to find the

connection between them, that is, to establish relations between the thermodynamically

defined tensions and moments, γ, ζ, B, Θ (Eq. 96), and the mechanical tensors of stresses and

moments, σ, M. This was done in Ref.[208] by direct calculation of δws in terms of purely

mechanical quantities. The following results were obtained [208]:

1 1 1 1 ;

2 2 2 2BH D BD H= + + Θ = + + Θγ σ ζ η (102)

1 2 1 21 1

( ) ; ( )2 2

= + = −σ σ σ η σ σ (103)

1 2 1 2 ; B M M M M= + Θ = − (104)

In the basis of principal curvatures the scalars σ1, σ2, M1, M2 are the eigenvalues of the

tensors σ and M, respectively. Thus, σ and η are isotropic and deviatoric tensions; B and Θ

are isotropic (bending) and deviatoric (torsion) moments.

What is remarkable in Eq. (102) is that the mechanical tensions, σ, η, do not coincide

with the thermodynamical ones, γ and ζ. In particular, this leads to ambiguity in defining

what is called "fluid interface". It could be either ζ = 0 (no work for shear), or η = 0 (isotropic

surface stresses) - see more extended discussion in Ref. [205].

In the case of a spherical surface of radius a we have B = 2M1 = 2M2 , D = 0, H = -1/a

and then Eqs. (95) and (102) yield

2 ;

22B B

P PI IIa a a= − − = +

γγ σ (105)

42

The second equation in (105) represents a form of the Laplace equation of capillarity. For the

so called surface of tension B = 0 by definition [143, 208]; then γ = σ and Eq. (105) coincides

with Eq. (95). On the other hand, if the Gibbs dividing surface is defined as the equimolecular

dividing surface (the surface for which the adsorption of solvent is equal to zero, see Ref. [8,

207], then B is not zero and the generalized Laplace equation should be used. It is interesting

to note that for a flat intermolecular dividing surface B is proportional to the distance, δ0,

between the surface of tension and the intermolecular dividing surface [209, 210]:

B0 = 2γ0 δ0 (106)

where the subscript "0" denotes flat interface (H = 0). Then by integrating Eq. (101) (with

γ = ωs) for a deformation of pure bending one obtains the Tolman equation [8, 9, 211]

γ γ γδ δ

= + + = − +

0 02

00 0

2

212

B H O Ha

Oa

( ) (107)

which expresses the physical dependence of surface tension on curvature. δ0 is often called

the Gibbs-Tolman parameter [9, 208].

C. Interfacial Bending Moment of Microemulsions

To illustrate the applicability of the above general expression let us consider a

specified system: a microemulsion containing spherical droplets (D = 0, cf. Eq. 97). A typical

microemulsion system contains the following components: water (w), oil (o), surfactant (s),

cosurfactant (c) and neutral electrolyte (e) - see Fig. 12. We choose the dividing surface to be

the equimolecular surface with respect to water, i.e. Γw = 0. Then at constant temperature T

and chemical potential of the oil phase, µ0, and for quasistatic processes without shear

deformation (γ = ωs, δβ = 0), Eq. (101) can be transformed to read [210]

( )s s s s c c e ed d d d BdH+ Γ = Γ − Γ − Γ +γ µ µ µ µ (108)

All differentials in the right-hand side of Eq. (108) are independent. Indeed, the number of

the independent intensive parameters in the system under consideration is equal to the number

of the components plus one [9]. Then from Eq. (108) one derives

, ,, , s c ec e

s

s H

BH

Γ

= Γ µ µµ µ

∂ µ∂∂ ∂

(109)

The integration of Eq. (109) yields

( ) ( )1, , , , ,s c e c e sB H B H BΓ = +µ µ µ µ (110)

43

where

0 , ,

s

s c e

ss sB d

H

Γ

Γ

= Γ

∫

µ µ

∂ µ∂

(111)

is the contribution of the surfactant to the bending moment.

The term B1(H,µc,µe) on the right-hand side of Eq. (110) represents the bending

moment of the an imaginary emulsion drop in a system containing only cosurfactant and

neutral electrolyte as solutes (Fig. 12b). For such a drop a counterpart of Eq. (108) holds:

( ) 1c c c c e ed d d B dH+ Γ = Γ − Γ +γ µ µ µ (112)

From Eq. (112) one derives

1

,, c ee

c

c H

BH

Γ

= Γ µµ

∂µ∂∂ ∂

(113)

By integrating Eq. (113) one obtains

( ) ( )1 2, , ,c e e cB H B H B= +µ µ µ (114)

Fig. 12. Illustration of the derivation of Eq. (118) for a water-in-oil microemulsion droplet: (a) water, oil, surfactant, cosurfactant and electrolyte; (b) water, oil, cosurfactant and electrolyte; (c) water, oil and electrolyte; (d) pure water and oil phases.

44

Here ( )

0 ,

c c

c e

cc cB d

H

Γ

Γ

= Γ

∫µ

µ

∂µ∂

(115)

represents the contribution of the cosurfactant to the interfacial bending moment. B2(H,µe) is

the bending moment of an imaginary emulsion drop in a system containing neutral electrolyte

of chemical potential µe in the aqueous phase (Fig. 12c). For such a system, instead of Eq.

112) one can write

( ) 2e e e ed d B dH+ Γ = Γ +γ µ µ (116)

Then by analogy with Eq. (114) one derives

( ) ( )2 , e p eB H B H B= +µ (117)

where ( )

0

e e

e

ee eB d

H

Γ

Γ

= Γ

∫µ ∂µ

∂

accounts for the contribution of the neutral electrolyte to the bending moment and Bp(H) is the

bending moment of an imaginary emulsion drop of surface curvature H in a system containing

only pure aqueous and oil phase (Fig. 12d). A combination of Eq. (110), (114) and (117)

leads to [210]

B B B B Bs c e p= + + + (118)

In other words, the total interfacial bending moment of a typical microemulsion droplet can

be represented as a superposition of four components.

Theoretical studies were devoted to calculation of the contributions of different

intermolecular interactions in B. The results [210] show that the contributions due to the

negative electrolyte adsorption (Be) and the dipole moments of the adsorbed molecules are

negligible compared to the contribution of the electric double layer and the van der Waals

interaction. In Ref. [212] the contribution of the van der Waals interaction to the value of B

for the phase boundary liquid-gas, as well as for the interface between pure water and

hydrocarbon phases was calculated. For a planar interface it was found that the van der Waals

contribution, Bvw, is of the order of 5 ×10−11 N, and tends to bend around the phase which has

larger Hamaker constant. Electrostatic effects, due to adsorption of ionic surfactants, were

shown to influence the bending moment considerably [210]. The Stern layer and the diffuse

part of the double electric layer provide comparable contributions, both in the range 1-

45

2 × 10−11 N, having the same sign as Bvw. In addition, the bending moment can be connected

with the experimentally measurable ∆V-potential across an interface [213], as well as with

steric effects due to the surfactant tails and headgroups, see Refs. [214-221] and Section E

below.

An approach to the calculation of γ and B is provided by the micromechanical theory,

which is related to the statistical mechanics. In the special case of spherical interface one may

use the expressions [208]:

0 0

2 2 2

2

1 2 2( ) , ( )

3 3

a r a rP r dr B P r dr

r a r a

∞ ∞

= ∆ + = ∆ −

∫ ∫γ (119)

where r is the radial coordinate and ∆P(r) ≡ PN −PT expresses the anisotropy of the pressure

tensor in the vicinity of the interface (r = a); PN and PT are the normal and tangential

components of the latter tensor with respect to the interface [9, 209]. Theoretical expressions

for ∆P(r) are available only for relatively simple systems. For example, supposing that only

van der Waals forces are operative, a model expression for ∆P(r) is derived [212] for liquid-

gas and oil-water interfaces. Fig. 13 represents the calculated curves ∆P vs r/a for benzene

droplet in air and benzene droplet in water. A pronounced difference between the curves for

liquid-gas and oil-water systems is seen: one maximum in the former case against maximum

and minimum in the latter case. The respective values of B, calculated from the curves in

Fig.13 by means of Eq. (119), are 3.5×10−11 N in the former case and 5.4×10−11 N in the latter

case.

Fig. 13. Anisotropy, ∆P, of the pressure tensor vs. r - a, where r is the radial distance and a = 100 nm is the droplet radius: Curve 1 - benzene drop in air; Curve 2 - benzene drop in water.

46

D. Model of Helfrich for the Surface Flexural Rheology

In view of Eq. (96) the work of flexural deformation of an interface can be expressed

in the form

dw B dH dDf = + Θ (120)

Sometimes Eq. (120) is written in the alternative (but equivalent) form [222-225]

dw C dH C dKf = +1 2

where K = H2 - D2 is the Gaussian curvature, and the coefficients C1 and C2 are algebraically

related to B and Θ [226]:

B C C H C D= + = −1 2 22 2, Θ

In general, B and Θ (as well as C1 and C2) are curvature dependent. To determine the

latter dependence one has to specify the flexural rheology of the interface. A frequently

employed model of Helfrich [202] assumes that the work of flexural deformation can be

written in the form

( ) KkHHkw ccf +−=2

02 (121)

where H0 is a parameter of the model called the spontaneous curvature, kc and kc are

coefficients (moduli) of bending and torsion elasticity, supposedly constants. From Eqs. (120)

and (121) one derives [205, 213, 226]

( ) DkHkkBB ccc 2,220 −=Θ++= (122)

where

B B k HH c0 0 04≡ = −=| (123)

is the bending moment of a planar interface. Note that Eq. (122) can be considered as

truncated series expansions of B and Θ for small curvatures (small deviations from planarity).

In addition, a comparison between Eqs. (106) and (123) shows that the quantities bending

moment, B0, spontaneous curvature, H0, and Gibbs-Tolman parameter, δ0, are closely related.

It should be noted also that the sign of the curvature H and the bending moment B is a

matter of convention, which is specified by the definition of the direction of the running unit

normal n to the surface. The general rule is that positive B tends to bend the interface around

the inner phase (the phase for which n is an outer normal) [213].

Combining Eqs. (104), (122) and (123) one can derive an expression for the

components of the tensor of the surface moments, M in the framework of the Helfrich model

[205]:

47

( )[ ] αβαβαβ bkaHkHkkM cccc −−+= 02 (124)

- see Section A above for the notation. The substitution of Eq. (124) into Eq. (93) yields the

form of the generalized Laplace equation in the framework of the Helfrich model [205]:

b k a H P Pc II Iαβαβ

αβαβσ − = −2 ,

(125)

It is interesting to note that kc does not take part in Eq. (125).

Eq. (125) is often applied to analyze capillary waves of small amplitude u, for which

Eq. (125) can be linearized; assuming isotropic surface tension, σ, from Eq. (125) one derives

[205]:

σ ∇ − ∇ ∇ = −s c s s II Iu k u P P2 2 2 (126)

where ∇s denotes surface gradient operator.

E. Physical importance of the bending moment and the curvature elastic moduli

In so far as the van der Waals, electrostatic and steric interactions can be treated as

being independent, they give additive contributions to B and Θ [213]:

B B B Bvw el st vw el st= + + = + +, Θ Θ Θ Θ (127)

Then in accordance with Eq. (122) one can seek B0, kc and kc in the form

B B B B k k k k k k k kvw el stc c

vwcel

cst

c cvw

cel

cst

0 0 0 0= + + = + + = + +, , (128)

The van der Waals contribution, Bvw0 , can be estimated by means of the expression

=

−= −+

2112

2

222

2

111

2

2/1

02

222

2

111

2

02

558

, ρραραραππ

γραραπ

HH

vw AA

B (129)

where γ0 is the surface tension of the pure water-air or water-oil interface, AH is the Hamaker

constant, ρ1 and ρ2 are the number densities of the two neighboring phases, αik are the

constants in the van der Waals potential: uik = −αik /r6; the subscripts "1" and "2" denote the

phase inside and outside the fluid particle, respectively. In general, Bvw0 tends to bend around

the phase, which has larger Hamaker constant [212]. For a oil-water interface Eq. (129)

predicts Bvw0 ≈ 5 × 10−11 N. Theoretical expressions for kc

vw and kcvw are not available in the

literature.

48

The contribution of the steric interaction can be related to the size and shape of the

tails and headgroups of the surfactant molecules [214-221]. For example, the following

expression was obtained [220] for such amphiphiles as the n-alkyl-poly(glycol-ethers),

(C2H4)n(OCH2CH2)mOH:

Bv b kT

ak

v bkT

ast

cst

M M0 4 5

22 2 2 3

4 641 12= − = +

π ε πε

~, ( ~ ) (130)

where ε% = (n – m)/(n + m) characterizes the asymmetry of the amphiphile, v is the volume of

an amphiphile molecule, aM is the interfacial area per molecule, k is the Boltzmann constant,

b is a molecular length scale in the self-consistent field model used [220].

Expressions for the electrostatic components of the curvature elastic moduli,

k kcel

celand , in terms of the nonlinear double layer theory can be found in Ref. [227].

Contributions of "point dipoles", diffuse and Stern parts of the electric double layer in B0 are

evaluated in Ref. [210]. Alternatively, one can relate Bel0 , k kc

elceland to the surface Volta

potential, ∆V, which is a directly measurable parameter [213]:

( ) ,2

18

2

0

+∆=

ds

VB elπε (131)

( ) ( )

++∆

++∆ −== 2

2

2

2331

24331

1222

,d

sds

Vd

kd

sds

Vd

k elc

elc π

επ

ε (132)

where ε is the dielectric constant, d is the distance between the positive and negative charges,

s is the distance between the surface charges and the Gibbs dividing surface (the position of

the latter is a matter of choice), see Fig. 14. ∆V in Eqs. (131) and (132) must be substituted in

CGSE-units, i.e. the value of ∆V in volts must be divided by 300. Note that ∆V expresses the

change of the surface potential due to the presence of an adsorption monolayer. ∆V can be

measured by means of the methods of the radioactive electrode or the vibrating electrode [10],

which give the drop of the electric potential across the interface.

Eqs. (131) and (132) can be used (i) when there is an adsorption layer of zwitterions or

dipoles, such as non-ionic and zwitterionic surfactants or lipids, at the interface; (ii) when the

electrolyte concentration is high enough and the counterions are located in a close vicinity of

the charged interface to form a "molecular capacitor"; (iii) when the surface potential is low:

then the Poisson-Boltzmann equation can be linearized and the diffuse layer behaves as a

molecular capacitor of thickness equal to the Debye screening length [228]. For example,

taking experimental value for zwitterionic lipids [229], ∆V = 350 mV, and assuming ε = 78.2,

49

d = 5Å, s/d << 1, from Eqs. (131) and (132) one calculates Bel0 =4.2×10−11 N, kc

el = 1.4×10−20

J and kcel = −0.7×10−20 J.

Despite of the small values of the above parameters, it turns out that the energy of

interfacial bending may considerably affect the droplet-droplet interactions in microemulsions

and emulsions [230]. Indeed, the collisions between droplets of low interfacial tension is often

accompanied by flattening of their surfaces in the zone of close contact, see Fig. 15, where the

radius of the formed transient flat film is denoted by r. By using Eqs. (121) and (123) one

may express the contribution of the bending energy to the droplet-droplet interaction, Wf ,

through the difference, ∆wf , between the droplet flexural energy after and before the film

formation [230]:

W r w r B H O H H rHf f= = − + <<( )2 2 1 12 2 20 0π π∆ / , ( ) (133)

Assuming a = 1/H = 300 nm, r ≈ a/50, |B0| = 5 ×10−11 N, and H/H0<<1, from Eq. (133) one

calculates |Wf| = 10 kT. In other words, the bending effects can be important for the

interaction between submicrometer emulsion droplets [230]. Since both the electrostatic and

van der Waals components of B0 tend to bend the interface around the oil phase, one can

expect that Wf < 0 (attraction) for aqueous droplets in oil, whereas Wf > 0 (repulsion) for oil

droplets in water. The existence of such an effect was established experimentally for water-in-

oil microemulsions [231]: the observed formation of doublets of droplets can be attributed to

the effect of Wf < 0.

Fig. 14. Sketch of a "molecular capacitor" of thickness d at a spherical interface, which can be formed either by adsorbed zwitterionic surfactant, or by charged surfactant headgroups and their counterions; the Gibbs dividing surface of radius a is chosen to be the boundary between the aqueous and the hydrophobic phases.

50

By means of similar considerations one can deduce [230] that an emulsion containing

microemulsion droplets in the continuous phase should be more stable than an emulsion

containing microemulsion droplets in the disperse phase, as it is observed experimentally

[232].

Recently, the effect of the interfacial bending moment (or the spontaneous curvature,

H0 = -B0/(4kc), see Eq. 123) was found to be important for the interaction between integral

proteins incorporated in lipid membranes [233].

V. THIN LIQUID FILMS STABILIZED BY SURFACTANTS

In addition to the previous sections (dealing with single interfaces) in the present

section we consider a couple of interfaces interacting across a thin liquid film. The presence

of surfactants has a crucial role for the properties of the thin liquid films, which are

constituent units of foams and some emulsions. Moreover, the adsorption of surfactants

modifies the surfaces of solid or fluid particles in the colloidal dispersions and thus strongly

affects the stability of the disperse systems. The presence of surfactant micelles also affects

the stability of colloids.

In this section we consider the properties of equilibrium thin liquid films. Section VI

is devoted to the surface forces of intermolecular origin acting in the liquid films or between

the particles in dispersions. Finally, in Sections VII - IX we describe the role of surfactants in

Fig. 15. Flat thin film of radius r and thickness h formed between two colliding fluid particles; the spherical part of the particle surface has radius a.

51

the hydrodynamic processes in liquid films and dispersions, which are coupled with

convective and diffusion transfer of surfactant.

A. Thermodynamic Description of a Thin Liquid Film

From a mathematical viewpoint a film is thin when its thickness is much smaller than

its dimension in lateral direction. From a physical viewpoint a liquid film formed between

two macroscopic phases is thin when the energy of interaction between the two phases across

the film is not negligible. The specific forces causing the interactions in a thin liquid film are

called surface forces [2-4].

In Fig. 16 a sketch of plane-parallel liquid film of thickness h is presented. The liquid

in the film exists in contact with the bulk liquid in the Plateau border. The film is

symmetrical, i.e. it is formed between two identical fluid particles (drops, bubbles) of internal

pressure P2. The more complex case of non-symmetrical and curved films is reviewed

elsewhere [143-234].

Fig. 16. The "detailed" and "membrane" models of a thin liquid film, on the left- and right-hand side, respectively.

Two different, but supplementary, approaches (models) are used in the macroscopic

description of a thin liquid film. The first of them, the "membrane approach", treats the film

as a membrane of zero thickness and film tension, γ, acting tangentially to the membrane - see

the right-hand side of Fig. 16. In the "detailed approach", the film is modeled as a

homogeneous liquid layer of thickness h and surface tension σf. The pressure P2 in the fluid

particles is larger than the pressure, P1, of the liquid in the Plateau border. The difference

Pc = P2 - P1 represents the capillary pressure of the liquid meniscus. By making the balance

of the forces acting on a plate of unit width along the y-axis and height h placed normally to

the film at −h/2 < z < h/2 (Fig. 16) one derives the Rusanov [235] equation:

52

γ σ= +2 fcP h (134)

Eq. (134) expresses a condition for equivalence between the membrane and detailed models

with respect to the lateral force. To derive the normal force balance one considers a parcel of

unit area from the film surface in the detailed approach. Since the pressure in the outer phase

P2 is larger than the pressure inside the liquid, P1, the mechanical equilibrium at the film

surface is ensured by the action of an additional disjoining pressure, Π(h) representing the

surface force per unit area of the film surfaces [236]

( )Π h P P Pc= − =2 1 (135)

- see the left hand side of Fig. 16. Note, that Eq. (135) is satisfied only at equilibrium; at non-

equilibrium conditions the viscous force can also contribute to the force balance per unit area

of the film. In general, the disjoining pressure, Π, depends on the film thickness, h. Note that

Π > 0 means repulsion between the film surfaces, whereas Π < 0 means attraction.

A typical Π(h)-isotherm is depicted in Fig. 17a. (The shape of the curve in Fig. 17a is

discussed in Section VI.A below.) One sees that the equilibrium condition, Π = Pc, can be

satisfied at three points shown in Fig. 17a. Point 1 corresponds to a film, which is stabilized

by the double layer repulsion; sometimes such a film is called the "primary film" or "common

black film". Point 3 corresponds to unstable equilibrium and cannot be observed

experimentally. Point 2 corresponds to a very thin film, which is stabilized by the short range

repulsion; such a film is called the "secondary film" or "Newton black film". Transitions from

common to Newton black films are often observed with foam or emulsion films [237-240].

In the framework of the membrane approach (Fig. 16) the film can be treated as a

single surface phase, whose Gibbs-Duhem equation reads [143, 241]:

d s d T dfi i

i

kγ µ= − −

=∑Γ

1 (136)

where γ is the film tension, T is temperature, sf is excess entropy per unit area of the film, Γi

and µi are the adsorption and the chemical potential of the i-th component. The Gibbs-Duhem

equations of phases 1 and 2 read

d P s d T n dii

k

iχ νχ χ µ χ= + =

=∑

11 2, , (137)

where sνχ and ni

χ are entropy and number of molecules per unit volume, and Pχ is pressure

(χ = 1, 2).

53

Fig. 17. (a) Sketch of a typical DLVO isotherm of the disjoining pressure, Π(h), and of the resulting ψ(h)-curve (b) calculated by means of Eq. (167).

Having in mind that Pc = P2 - P1 from Eq. (137) one can derive an expressions for dPc.

Multiplying this expression by h and subtracting the result from the Gibbs-Duhem equation of

the film, Eq. (136) one obtains [241]

d s d T hd P dc i ii

kγ µ= − + −

=∑~ ~Γ

1 (138)

where

( ) ( )~ , ~ , ,...,s s s s h n n h i kfi i i i= + − = + − =ν ν

2 1 2 1 1Γ Γ (139)

To specify the model one needs an additional equation to determine the multiplier h. For

example, let this equation be

1 0Γ =% (140)

54

Eq. (140), in view of Eq. (139), requires h to be the thickness of a liquid layer from phase 1,

containing the same amount of component 1 as the real film [242, 243]. This thickness is

called the thermodynamic thickness of the film [243]. It can be of the order of the real film

thickness if component 1 is chosen in an appropriate way, say the solvent in the film phase.

Combining Eqs. (135), (136) and (140) one obtains [242]

2

k

i ii

d sdT hd d=

= − + Π − Γ∑ %%γ µ (141)

A corollary of Eq. (141) is the Frumkin [244] equation

2, ,..., kT

h

= Π µ µ

∂γ∂

(142)

Eq. (142) predicts a rather weak dependence of the film tension γ on the disjoining pressure,

Π, in equilibrium thin films (small h). The substitution of Eqs. (134) and (135) into Eq. (141)

yields [243]

22

kf

i ii

d sdT dh d=

= − − Π − Γ∑ %%σ µ (143)

From Eq. (143) one can derive the following useful relations [242]

2, ,...,

2k

f

Th

= −Π µ µ

∂σ∂

(144)

( ) ( )12

f l

h

h h dh∞

= + Π∫σ σ (145)

with σl being the surface tension of the bulk liquid. In particular, Eq. (145) allows calculation

of the film surface tension (and the film tension) when the disjoining pressure isotherm, Π(h),

is known.

The above thermodynamic equations are in fact corollaries from the Gibbs-Duhem

equation of the membrane approach Eq. (136). There is an equivalent and complementary

approach, which treats the two film surfaces as separate surface phases with their own

fundamental equations [235, 242, 245], thus for a flat symmetric film one postulates.

12 ,

kf f f f

i ii

dU TdS dA dN Adh=

= + + − Π∑σ µ (146)

where A is area; Uf, Sf and Nif are excess internal energy, entropy and number of molecules

ascribed to the film surfaces. Compared with the fundamental equation of a simple surface

phase [9] Eq. (146) contains an additional term, ΠAdh, which takes into account the

55

dependence of the film surface energy on the film thickness. Eq. (146) provides an alternative

thermodynamic definition of the disjoining pressure:

1 fUA h

Π = −

∂∂

(147)

B. Contact Line, Contact Angle and Line Tension

B.1. Force balance at the contact line

The thin liquid films formed in foams or emulsions exist in a permanent contact with

the bulk liquid in the Plateau border, encircling the film. From a macroscopic viewpoint, the

boundary between film and Plateau border is treated as a three-phase contact line: the line, at

which the two surfaces of the Plateau border (the two concave menisci sketched in Fig. 16)

intersect at the plane of the film - see the right-hand side of Fig. 16. The angle, θ0, subtended

between the two meniscus surfaces, represents the thin film contact angle.

The interactions between the neighboring phases in a close vicinity of the contact line

lead to the appearance of excess energy per unit length of the contact line. The latter is

ascribed to the contact line as an effective line tension, κ, acting tangentially to the line [8].

For a curved contact line of radius rc the line tension gives rise to a contribution of magnitude

σκ = κ/rc in the balance of the surface tensions at the contact line [246-251]. As a force (per

unit length) σκ is directed toward the center of curvature of the contact line. Thus, the force

balance at the periphery of a symmetrical flat film with circular contact line, like those

depicted in Fig. 16, reads [252]

γ κ σ θ+ =rc

l2 0cos , (148)

There are two film surfaces and two contact lines in the detailed approach - see the

left-hand side of Fig. 16. They can be treated thermodynamically as linear phases and an one-

dimensional counterpart of Eq. (146) can be postulated [252]:

d U T d S d L d N d hL Li i

L

i= + + +∑2 ~κ µ τ (149)

Here UL, SL and NiL are linear excesses, ~κ is the line tension in the detailed approach and

τ∂∂

=

1L

Uh

L (150)

56

is an one-dimensional counterpart of the disjoining pressure - cf. Eq. (147). The quantity τ,

called the transversal tension, takes into account the interaction between the two contact

lines. For the case of a flat symmetric film the tangential and normal projections of the force

balance at the contact lines read (see the left-hand side of Fig. 16) [252]:

σ κ σ θf

c

lr

+ =~

cos1

(151)

τ σ θ= l sin (152)

Note that in general θ ≠ θ0 - see Fig. 16. Besides, both θ0 and θ can depend on the radius of

the contact line due to line tension effects. In the case of straight contact line from Eqs. (145)

and (151) one derives [243]

( )cosθ σσ σrc

f

l lh

h d h1

1 12=∞

∞

= = + ∫ Π (153)

Since cosθ ≤ 1, the surface tension of the film must be less than the surface tension of the

bulk solution, σf < σl, and the integral term in Eq. (153) must be negative in order for a

nonzero contact angle to be formed. Hence, the contact angle, θ, and the transversal tension, τ

(cf. Eq. 152), are integral effects of the long-range attractive surface forces acting in the

transition zone between the film and Plateau border, where h > h1 - see Fig. 17a.

In the case of a fluid particle attached to a surface (Fig. 18) the integral of the pressure

P1(z) = P1(z=0) − ∆ρgz over the particle surface equals the buoyancy force, Fb, which at

equilibrium is counterbalanced by the disjoining pressure and transversal tension forces [253]

2 1 12π τ πr F rc b c= + Π (154)

Fb is negligible for bubbles of diameter smaller than c.a. 300 µm. Then the forces due to τ and

Π counterbalance each other. Hence, at equilibrium the role of the repulsive disjoining

pressure is to keep the film thickness uniform, whereas the role of the attractive transversal

tension is to keep the bubble (droplet) attached to the surface. In other words, the particle

sticks to the surface at the contact line where the long-range attraction prevails (see Fig. 17a),

whereas the repulsion predominates inside the film, where Π = Pc > 0. Note that this

conclusion is valid not only for particle-wall attachment, but also for particle-particle

interaction. For zero contact angle τ is also zero (Eq. 152) and the particle will rebound from

the surface (the other particle), unless some additional external force does keep it attached.

The deeper understanding of such phenomena has not only fundamental, but also practical

57

importance for phenomena like flocculation in emulsions, or redeposition of oil droplets on

solid surfaces in washing.

Fig. 18. Sketch of the forces exerted on a fluid particle (bubble, drop, vesicle) attached to a planar surface: Π is disjoining pressure, τ is transversal tension, P1 is the pressure in the outer liquid phase.

B.2. Transition zone film - Plateau border

Macroscopic parameters like contact angle, θ, line and transversal tensions, κ and τ,

can be deduced from the shape of the microscopic transition zone between the film and the

Plateau border. From a microscopic viewpoint the transition between the film surface and the

meniscus is smooth as depicted in Fig. 19 (the continuous lines). As the film thickness

increases across the transition zone, the disjoining pressure decreases and tends to zero for h

» 10 nm, cf. Fig. 17a and Fig. 19. Respectively, the surface tension varies from σf for the film

to σl for the Plateau border. By using local force balance considerations one can derive the

equations governing the shape of the meniscus in the transition zone; in the case of axial

symmetry, depicted in Fig. 19, these equations read [252]

( ) ( ) ( ) ( )1sin sin cd r r P rdr r

+ = − Πσ ϕ σ ϕ (155)

( ) ( ) ( ) ( )1cos sin , tancd dzr r P rdz r dr

− + = =σ ϕ σ ϕ ϕ (156)

where ϕ(r) is the running slope angle. Equations (155)-(156) allow calculation of the three

unknown functions, z(r), ϕ(r) and σ(r), provided that the disjoining pressure, Π(r), is known

from the microscopic theory. For example, Π(r) may be approximately expressed through the

58

disjoining pressure Π(h) of a plane-parallel film of thickness h ≡ 2z(r), [254, 255]. By

eliminating Pc between Eqs. (155) and (156) one can derive [252]

( ) ( )cosd r rdz

= − Πσ ϕ (157)

This result shows that the hydrostatic equilibrium in the transition region is ensured by

simultaneous variation of σ and Π. Eq. (157) represents a generalization of Eq. (144) for a

film of uneven thickness and axial symmetry. Generalization of Eqs. (155) - (156) for the case

of more complicated geometry is also available [234].

In the Plateau border z » h, Π → 0, σ → σl = const and both Eqs. (155) and (156)

reduce to the common Laplace equation of capillarity expressed in the following parametric

form [256]:

ddr r

Pclsin sin /ϕ ϕ σ+ = (158)

The macroscopic contact angle, θ, is defined as the angle at which the extrapolated meniscus,

obeying Eq. (158), meets the extrapolated film surface [251] - see the dashed line in Fig. 19.

The real surface, shown by solid line in Fig. 19, differs from this extrapolated (idealized)

profile, because of the interactions between the two film surfaces, which is taken into account

in Eq. (155), but not in Eq. (158). To compensate for the difference between the real and

Fig. 19. Liquid film of thickness h: The solid lines represent the real interfaces, whereas the dashed lines show the extrapolated interfaces in the transition zone film -Plateau border; θ is the contact angle and ϕ(r) is the running slope angle.

59

idealized system, line and transversal tensions are ascribed to the contact line in the

macroscopic approach. In particular, the line tension makes up for the differences in surface

tension and running slope angle [252]

~ sincos

sincos

;real idealized

κ σ ϕϕ

σ ϕϕr r r

d rc

rB=

−

∫

2 2

0 (159)

whereas τ compensates for the differences in surface forces (disjoining pressure):

( ) ( )[ ]τ = −∫1

0rr r d r

c

rB Π Πid (160)

where

( )( )

id

id

for 0 ;

0 for .c c

c

P r r

r r

Π = < <

Π = >

The superscripts "real" and "idealized" in Eq. (159) mean that the quantities in the respective

parentheses must be calculated for the real and idealized meniscus profiles; the latter coincide

for r > rB - cf. Fig. 19.

Theoretical calculations based on the above micromechanical theory predict κ ≈

−10−12 N for foam films [257] and κ ≈ −10−13 N for emulsion films [258]. Correspondingly,

the experiments [259] with foam films stabilized by sodium dodecyl sulfate (SDS) show that

the equilibrium value of κ is zero in the framework of the experimental accuracy, ±5×10-9 N.

Therefore it seems that the equilibrium line tension is a very small quantity and can be

neglected in most cases (except may be in heterogeneous nucleation). On the other hand,

comparatively large non-equilibrium values of the line tension, κ ≈ −10-7 N, have been

measured with advancing menisci (shrinking bubbles). Such large values can be explained

[260] by the strong short-range attraction between the two film surfaces in the case of

secondary films stabilized by SDS [261-263]. The detachment of the two film surfaces in the

process of meniscus advance is accompanied by significant local alterations of the interfacial

shapes and tensions in the real dynamic transition zone, which contributes (cf. Eq. 159) to

large values of the dynamic line tension. Moreover, if one carries out the same experiment,

but with a nonionic surfactant, rather than with SDS, no detectable line tension is found out

[264]. This is related to the absence of strong short-range attraction between the two nonionic

surfactant adsorption monolayers, as indicated by the small contact angle which is not

subjected to hysteresis in this case [264].

60

In conclusion, it should be noted that the width of the transition region between a thin

liquid film and Plateau border is usually very small [257]- below 1 µm. That is the reason

why the optical measurements of the meniscus profile give information about the thickness of

the Plateau border in the region r > rB (Fig. 19). Then if the data are processed by means of

Laplace equation, Eq. (158), one determines the contact angle, θ, as discussed above. In spite

of being a purely macroscopic quantity, θ characterizes the magnitude of the surface forces

inside the thin liquid film, as implied by Eq. (153). This has been first pointed out by

Derjaguin [265] and Princen and Mason [266].

C. Contact Angle and Interaction between Surfactant Adsorption Monolayers

Equation (153) can be written in the form [245]

f f h dhh

= − − ≡∞

∫2 1 00

σ θ( cos ), ( )Π (161)

where f is the free energy (per unit area) of interaction between the two film surfaces and σ ≡

σl. Indeed, the integral in Eq. (161) expresses the work for bringing the two film surfaces

from infinity to a finite distance h0. Therefore, the contact angle measurements provide a

method for determining the interaction between two adsorption monolayers of amphiphilic

molecules such as various surfactants, lipids, proteins. (See Ref. [267] for a review on the

methods for thin film contact angle measurements.) In this aspect, the method based on

measurement of contact angles is alternative and complementary to the surface force

apparatus, which detects the force between two crossed mica cylinders [4]. The advantage of

the contact angle measurements is that the experimental system is relatively simple, self-

adjusting, inexpensive, and allows measurements with fluid interfaces. A drawback of the

method is that it allows measurements only in a restricted range(s) of film thicknesses,

corresponding to ∂Π/∂h < 0, see e.g. Ref. [268].

C.1. Hysteresis of the contact angle

In the case when the equilibrium film is secondary (h = h2 in Fig. 17a), hysteresis of

the contact angle is often observed. This allows one to measure three experimental parameters

(instead of one): the equilibrium contact angle, θ0, the advancing contact angle, θA, and the

61

receding contact angle, θR. In particular, θA corresponds to detachment of the two film

surfaces, while θR corresponds to attachment of the film surfaces. For example, the following

values have been measured [269] with foam films stabilized by the protein lyzozyme at

pH = 11.5: θR = 2.3o, θ0 = 3.2o, θA = 16.4o. Contact angle hysteresis has been observed also

with foam films stabilized with sodium dodecyl sulfate [270]. These are examples for contact

angle hysteresis observed with molecularly smooth and homogeneous surfaces; in this case

the hysteresis is due to the action of the surface forces in the film [271].

In general, θR ≤ θ0 ≤ θA. The difference between the contact angle hysteresis with

liquid films and with solid surfaces is that in the former case an equilibrium angle does exist,

whereas in the latter case there is no specified equilibrium contact angle in the range between

θR and θA. For equilibrium primary liquid films (h = h1 in Fig. 17a) the values of the three

angles usually coincide in the framework of the experimental accuracy.

In principle, the experimental data about θR, θ0, and θA allow one to extract

information about the shape of the Π(h)-isotherm, cf. Fig. 17a, by using an appropriate model

Π(h)-curve depending on several parameters, which are to be determined from the

comparison with the experiment, see e.g. Ref. [254, 255]. The measured value of θ0 can be

interpreted by means of Eq. (161), were h0 is the equilibrium thickness of the film. The values

of θA and θR can be interpreted on the basis of the theory by Martynov et al. [271, 272],

which is briefly outlined below.

Let us consider a planar thin liquid film in contact with a Plateau border. The capillary

pressure Pc = P2 - P1 can be varied by ejection or injection of liquid through the orifice in the

wall of the Scheludko cell [273], see Fig. 20. Such a process is accompanied by receding or

advance of the capillary meniscus.

For the sake of simplicity we consider a Plateau border of translational symmetry

along the y-axis, Fig. 20. In addition, we assume that the vertical wall is well wettable by the

liquid, i.e. dz/dx →∞ at the wall. The last two assumptions do not restrict the validity of the

final expressions for θR, θ0, and θA in so far as the dependence of the contact angle on the

capillary pressure and the curvature of the contact line is negligible.

62

Fig. 20. Sketch of the transition zone film - Plateau border in a Scheludko cell; h is the film thickness, θ is the contact angle subtended between the extrapolated meniscus (the dashed line) and the film midplane.

The meniscus profile z(x) obeys a counterpart of Eq. (155):

( ) )('1

''232

hPzz

c Π−=+

σ (162)

where the prime denotes d/dx, and h = 2z(x). The variation of σ along the transition zone is

small (∆σ/σ < 1%) and is neglected in the present derivation. The extrapolated meniscus

profile, ~z (x), (the dashed line in Fig. 20) obeys a counterpart of Eq. (158),

( ) cPzz

=+

232'~1''~σ

(163)

which follows from Eq. (162) for Π = 0. The respective boundary conditions are ~' ( ) tan ; ( ) ~( ) ; ' ( ) ~' ( )z r z R z R H z R z Rc = = = = = ∞θ (164)

- see Fig. 20. Multiplying Eq. (163) by ~'z and integrating form ~z = 0 to ~z = H one obtains

HPc =σ θcos (165)

A similar integration of Eq. (162) from z = h/2 to z = H yields [271, 272] 2

1 2σ ψ

+=

zh Pc

'( , ) (166)

where

63

∫∞

Π−−≡hcc dhhPhHPh )()2(),(ψ (167)

At equilibrium the meniscus surface levels off at the planar film surface, where z' = 0.

Setting z' = 0 in Eq. (166), in keeping with Eq. (167) one obtains

∫∞Π−

−=

0

)(2

122 0

0 hdhh

H

hHPσ (168)

where P0 and h0 are the values of Pc and h corresponding to the equilibrium state. Neglecting

the term h0/2H in the parenthesis and substituting HP0 from Eq. (165) one arrives at the

expression for the equilibrium contact angle,

∫∞Π+=

0

)(211cos 0 h

dhhσ

θ (169)

cf. Eq. (153). The differentiation of Eq. (167) yields

∂ψ∂h

Pc= −Π (170)

The comparison of the last equation with Eq. (135) shows that the dependence ψ vs. h has

extrema in the points h = h1, h2, h3, see Fig. 17b. Moreover, for the equilibrium film one has

ψ = 2σ (set z' = 0 in Eq. 166) - see the horizontal dashed line in Fig. 17b, corresponding to

the stable secondary film with h = h2.

C.2. Expression for the advancing contact angle

Fig. 21 represents a close vicinity of the contact zone film - Plateau border; the small

rectangles symbolize the amphiphilic molecules of the two adsorption monolayers. The

peripheral molecules of the film experience a normal force per unit length, σsinθcr , which

tends to detach the two film surfaces. In fact, θcr denotes the critical value of the microscopic

contact angle corresponding to the onset of detachment of the peripheral molecules and the

beginning of the meniscus advance. The respective values of the capillary pressure are Pc =

PA and h = hA. The angle θcr is to be distinguished from the macroscopic advancing contact

angle, θA, which is related to the slope of the extrapolated meniscus surface at the film

midplane, see Fig. 21.

Applying Eqs.(165)-(167), with z'(rc) = tanθcr, to the advancing meniscus one derives

A A AcosH P = σ θ (171)

( )A

cr A A A A2 cos 2 1 / 2 ( )h

H P h H h dh∞

= − − Π∫σ θ (172)

64

Fig. 21. Sketch of a secondary film representing two attached monolayers of amphiphile molecules (the rectangles); σsinθcr is the critical force per unit length of the film periphery causing detachment of the peripheral molecules.

Neglecting the second term in the parenthesis and combining Eqs. (171) and (172) one

obtains an expression for the advancing contact angle:

AA cr

1cos cos ( )2 h

h dh∞

= + Π∫θ θσ

(173)

The expression proposed by Martynov et al. [271] corresponds to setting θcr = π/2 in Eq.

(173). The first term in the right-hand side of Eq. (173) expresses the contribution of the

contact adhesion, whereas the integral term expresses the contribution of the long-range

interaction. Comparing Eqs. (169) and (173) one obtains

A

00 A cr

1cos cos 1 cos ( )2

h

hh dh− = − + Π∫θ θ θ

σ (174)

As the Π(h)-isotherm is rather steep for secondary films, one may expect that hA ≈ h0 and the

integral term in Eq. (174) can be neglected. If such is the case, it turns out that the difference

between the equilibrium and the advancing contact angles is determined entirely by cosθcr,

i.e. by the contact adhesion.

The depth of the primary minimum, Πmin , may be estimated from the value of the

advancing contact angle as follows. Since Π is force per unit area, then Πa (with a being area

per adsorbed amphiphile molecule) gives force per molecule. Then the critical force, fcr ,

needed to detach two adhered amphiphile molecules can be identified with

65

cr minf a= Π (175)

On the other hand, from Fig. 21 one realizes that cr crsinf a= σ θ . Then in view of Eq. (175)

one obtains

min crsina

Π =σ θ (176)

With the aforementioned values measured with lysozyme, θ0 = 3.2o, θA = 16.4o, √a=3.4 nm

and by means of Eqs. (174) and (176) one estimates θcr=16.08o and Πmin = − 4.4×107 dyn/cm2.

One may define wcr = 2σ(1 - cosθcr) as the critical work of detachment of the two

adsorption monolayers. With the above value of θcr and σ = 54 dyn/cm one calculates

wcr = 2.1 erg/cm2 for two attached lysozyme monolayers.

C.3. Expression for the receding contact angle

Imagine an equilibrium initial state of the film, corresponding to Pc = P0 , h = h2 and

ψ = 2σ, see Fig. 17b. Even an infinitesimal increase of Pc will break the equilibrium condition

ψ = 2σ, cf. Eq. (167), and the ψ vs. h curve will move upwards (see the dotted line in Fig

17b). This will cause a motion (receding) of the meniscus, accompanied by attachment of the

two meniscus surfaces [271]. From Eq. (166), transformed to read

( )[ ] 212 1/2' −= ψσz , (177)

one sees that the domain of film thicknesses, for which ψ > 2σ, is unstable, because z' in Eq.

(177) takes imaginary values. More precisely, the viscous forces, not accounted for in Eq.

(177), affect the meniscus shape in this narrow region.

At some Pc = PR the maximum at h = h1 (see the dotted line in Fig. 17b) will touch the

line ψ = 2σ. Then for any Pc > PR the region around h = h1 becomes also unstable. Thus the

meniscus motion responses much more easily to the applied increased capillary pressure. As

pointed out by Martynov et al. [271], the receding of the meniscus takes place at the critical

conditions, Pc = PR and θ = θR, with θR being the receding contact angle. As shown in Fig

17b, the maximum of the dotted ψ(h) curve, which touches the line ψ = 2σ, corresponds to a

metastable primary film. From a physical viewpoint this means that a thicker primary film

66

will be left after the receding contact line. In fact, this is what is observed experimentally: a

narrow belt of greater thickness is present in an intermediate vicinity behind the receding

meniscus [269, 274, 275]. Applying Eqs.(165)-(167) to the receding meniscus (H = HR,

Pc = PR) one derives [271]

R R RcosH P = σ θ (178)

( )R

R R R R2 2 1 / 2 ( )h

H P h H h dh∞

= − − Π∫σ (179)

where hR denotes the thickness of the film (corresponding to the maximum of the dotted line

with ψ = 2σ in Fig. 17b) left after the receding Plateau border. Neglecting the second term in

the parenthesis in Eq. (179), and combining it with Eq. (178), one obtains an expression for

the receding contact angle:

RR

1cos 1 ( )2 h

h dh∞

= + Π∫θσ

(180)

The comparison between Eqs. (169) and (180) yields

0

R

0 R1cos cos ( )

2h

hh dh− = Π∫θ θ

σ (181)

For Π(h) isotherms like that depicted in Fig. 17a the integral in the right-hand side of Eq.

(181) is typically negative; this implies that θ0 > θR , which is the usual experimental

situation. Eq. (181) shows that θR brings information about the shape of the Π(h)-isotherm in

the region of the loop (h2 < h <hR ≈ h1), cf. Fig. 17a. One can extract this information by using

an appropriate model Π(h)-curve depending on several parameters, which are to be

determined from the comparison with the experiment.

D. Films of Uneven Thickness

The gap between two colliding particles (bubbles, droplets, solid particles, surfactant

micelles) in a colloidal dispersion can be treated as a film of uneven thickness. Then it is

possible to utilize the theory of thin films to calculate the energy of interaction between two

colloidal particles. Derjaguin [276] has derived an approximate formula, which expresses the

energy of interaction between two spherical particles of radii R1 and R2 through integral of

the excess surface free energy per unit area, f(h), of a plane-parallel film of thickness h (see

Eq. (161) above):

67

( ) ( )U hR R

R Rf h dh

h0

1 2

1 2

2

0

=+

∞

∫π

(182)

Here h0 is the shortest distance between the surfaces of the two particles and U(h0) is the

respective particle-particle interaction energy, see Fig. 22. In the derivation of Eq. (182) it is

assumed that the interaction between two parcels from the particle surfaces, separated at the

distance h, is approximately the same as that between two similar parcels in a plane-parallel

film. This assumption is correct when the range of action of the surface forces and the

distance h0 are small compared to the curvature radii R1 and R2. It has been established, both

experimentally [4] and theoretically [277], that Eq. (182) provides a good approximation in

the range of its validity.

Fig. 22. Two spherical particles of radii R1 and R2; h0 and h are the shortest and the running surface-to-surface distance, respectively.

Eq. (182) can be generalized for smooth surfaces of arbitrary shape (not necessarily

spheres). For that purpose the surfaces of the two particles are approximated with paraboloids

in the vicinity of the point of closest approach (h = h0). Let the principal curvatures at this

point be c1 and c1′ for the first particle and c2 and c2

′ for the second particle. Then the

generalization of Eq. (182) reads [2]

( ) ( )U hC

f h dhh

02

0

=∞

∫π , (183)

( ) ( )2 21 1 2 2 1 2 1 2 1 2 1 2sin cosC c c c c c c c c c c c cω ω′ ′ ′ ′ ′ ′≡ + + + + +

68

where ω is the angle subtended between the directions of the principle curvatures of the two

approaching surfaces. For two spheres one has c c R c c R1 1 1 2 2 21 1= ′ = = ′ =/ , / and Eq.

(183) reduces to Eq. (182).

For two cylinders of radii r1 and r2 crossed at angle ω one has c1 = c2 = 0; c r1 11′ = / ,

c r2 21′ = / and Eq. (182) yields

( ) ( )U hr r

f h dhh

01 22

0

=∞

∫π

ωsin (184)

Eq. (184) is often used in connection to the experiments with the surface force apparatus [4,

278], where the surfaces represent two crossed cylindrical mica sheets. The divergence in Eq.

(184) for ω = 0 is related to the fact that the axes of the two infinitely long cylinders are

parallel for ω = 0 and thus the area of the zone of interaction becomes infinite. Eq. (184) can

be also utilized to calculate the energy of interaction between two cylindrical surfactant

micelles.

The Derjaguin approximation is applicable to any type of force law (attractive,

repulsive, oscillatory) if the range of the forces is much smaller than the particles radii. It

reduces the problem for interactions between particles to the simpler problem for interactions

in plane-parallel films, which are reviewed in the next section.

(continued in Part 2)

Part 2 of this review and the List of References can be found at:

http://www.lcpe.uni-sofia.bg/publications/1999/pdf/1999-10-2-KD-PK-II.pdf